10895:
235:
8275:
9679:
3069:
7585:
9691:
36:
135:
8270:{\displaystyle {\begin{aligned}z^{1}-1&=z-1\\z^{2}-1&=(z-1)(z+1)\\z^{3}-1&=(z-1)(z^{2}+z+1)\\z^{4}-1&=(z-1)(z+1)(z^{2}+1)\\z^{5}-1&=(z-1)(z^{4}+z^{3}+z^{2}+z+1)\\z^{6}-1&=(z-1)(z+1)(z^{2}+z+1)(z^{2}-z+1)\\z^{7}-1&=(z-1)(z^{6}+z^{5}+z^{4}+z^{3}+z^{2}+z+1)\\z^{8}-1&=(z-1)(z+1)(z^{2}+1)(z^{4}+1)\\\end{aligned}}}
8924:. This by itself doesn't prove the 105th polynomial has another coefficient, but does show it is the first one which even has a chance of working (and then a computation of the coefficients shows it does). A theorem of Schur says that there are cyclotomic polynomials with coefficients arbitrarily large in
8468:
3496:
469:
3344:
6779:
6376:
4754:
1989:
5431:
5752:
7054:
4019:
is not convenient, because it contains non-primitive roots, such as 1, which are not roots of the cyclotomic polynomial, and because it does not give the real and imaginary parts separately.) This means that, for each positive integer
654:
5870:
5597:
8290:
6630:
6193:
2178:
6272:
1196:
4609:
8857:
4873:
328:
5091:
3355:
5004:
4369:
3185:
7294:
3578:
2927:
6907:
3785:
9609:
9483:
976:
870:
7563:
7590:
4680:
9044:
5010:
runs over the roots of the above polynomial. As for every cubic polynomial, these roots may be expressed in terms of square and cube roots. However, as these three roots are all real, this is
7453:
6056:
9372:
2308:
9540:
2752:
1690:
8985:
1780:
3706:
2981:
9100:
2868:
2833:
2790:
2651:
2614:
2526:
1488:
3224:
1536:
9163:
th roots of unity with the additional property that every value of the expression obtained by choosing values of the radicals (for example, signs of square roots) is a primitive
6287:
4139:
4941:
1870:
5300:
4308:
3058:
3876:
9413:
9305:
5627:
4278:
2680:
1645:
7172:
4774:
2556:
1579:
4024:, there exists an expression built from integers by root extractions, additions, subtractions, multiplications, and divisions (and nothing else), such that the primitive
1441:
4017:
6669:
1818:
303:
4398:
4227:
4183:
4532:
559:
8747:
5787:
4802:
8463:{\displaystyle \Phi _{n}(z)=\prod _{d\,|\,n}\left(z^{\frac {n}{d}}-1\right)^{\mu (d)}=\prod _{d\,|\,n}\left(z^{d}-1\right)^{\mu \left({\frac {n}{d}}\right)},}
5505:
5043:
10715:
7367:
over the rational numbers (that is, it cannot be written as the product of two positive-degree polynomials with rational coefficients). The case of prime
6513:
6947:
4946:
695:
Subsequent sections of this article will comply with complex roots of unity. For the case of roots of unity in fields of nonzero characteristic, see
6125:
2110:
6204:
1051:
9264:
with periodic boundaries), the orthogonality property immediately follows from the usual orthogonality of eigenvectors of
Hermitian matrices.
10597:
10521:
10288:
10185:
10158:
10120:
10039:
10010:
4056:
3491:{\displaystyle \left(\cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}\right)^{\!k}=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}}\neq 1}
4313:
3096:
7208:
3518:
2883:
4799:, there are two primitive sixth roots of unity, which are the negatives (and also the square roots) of the two primitive cube roots:
10684:
10654:
10632:
10386:
10250:
10215:
10081:
9686:, the red points are the fifth roots of unity, and the black points are the sums of a fifth root of unity and its complex conjugate.
3008:
3002:
119:
6837:
10589:
3725:
464:{\displaystyle \exp \left({\frac {2k\pi i}{n}}\right)=\cos {\frac {2k\pi }{n}}+i\sin {\frac {2k\pi }{n}},\qquad k=0,1,\dots ,n-1.}
10344:
10789:
4885:. The sum of a root and its conjugate is twice its real part. These three sums are the three real roots of the cubic polynomial
10708:
9547:
9421:
901:
10403:
9956:
8889:. It is not a surprise it takes this long to get an example, because the behavior of the coefficients depends not so much on
774:
57:
53:
100:
7490:
10513:
72:
8990:
7569:
9666:
9120:
Many restrictions are known about the values that cyclotomic polynomials can assume at integer values. For example, if
3990:
can be used to show that the cyclotomic polynomials may be conveniently solved in terms of radicals. (The trivial form
10803:
10546:
9645:
9249:
8920:
for which there could be a coefficient besides 0, 1, or −1 is a product of the three smallest odd primes, and that is
7068:
7381:
7346:
6278:
6011:
3960:
2046:
79:
10924:
10919:
10701:
9317:
6913:
5758:
2255:
212:
185:
9660:
abelian extension of the rationals is such a subfield of a cyclotomic field – this is the content of a theorem of
4028:
th roots of unity are exactly the set of values that can be obtained by choosing values for the root extractions (
46:
9507:
9207:
7372:
196:
2724:
1650:
10939:
8931:
4412:-gon is constructible with compass and straightedge. Otherwise, it is solvable in radicals, but one are in the
1732:
506:
86:
9976:
275:
3339:{\displaystyle \left(\cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}\right)^{\!n}=\cos 2\pi +i\sin 2\pi =1,}
9159:, as roots of unity are themselves radicals. Moreover, there exist more informative radical expressions for
6371:{\displaystyle \operatorname {SP} (n)=\sum _{d\,|\,n}\mu (d)\operatorname {SR} \left({\frac {n}{d}}\right).}
4749:{\displaystyle {\frac {\varepsilon {\sqrt {5}}-1}{4}}\pm i{\frac {\sqrt {10+2\varepsilon {\sqrt {5}}}}{4}},}
3651:
2952:
1821:
9049:
2842:
2807:
2764:
2625:
2588:
2500:
1448:
10642:
9996:
9845:
9494:
7364:
7182:
7087:
5237:
3915:
3076:
3060:
The roots of the minimal polynomial are just twice the real part; these roots form a cyclic Galois group.
2324:
1499:
1376:
876:
704:
520:
68:
8873:. Thus a necessary (but not sufficient) condition for a repunit to be prime is that its length be prime.
3931:
th roots of unity are roots of an irreducible polynomial (over the integers) of lower degree, called the
1984:{\displaystyle a={\frac {\operatorname {lcm} (k,n)}{k}}={\frac {kn}{k\gcd(k,n)}}={\frac {n}{\gcd(k,n)}}.}
10750:
7191:
7114:
5426:{\displaystyle x_{j}=\sum _{k}X_{k}\cdot z^{k\cdot j}=X_{1}z^{1\cdot j}+\cdots +X_{n}\cdot z^{n\cdot j}}
4155:
4105:
3938:
3629:
2364:
1853:
10894:
4888:
234:
5747:{\displaystyle x_{j}=\sum _{k}A_{k}\cos {\frac {2\pi jk}{n}}+\sum _{k}B_{k}\sin {\frac {2\pi jk}{n}}.}
3633:
10929:
10745:
10458:
10374:
9785:, the primitive roots of unity are not quadratic integers, but the sum of any root of unity with its
9650:
9222:
9203:
9192:
9168:
8894:
7064:
6917:
6813:
4526:
4283:
3897:
3879:
3014:
2874:
314:
5814:
4400:
is an irreducible polynomial whose roots are all real. Its degree is a power of two, if and only if
3835:
10934:
9961:
9393:
9375:
9285:
8281:
5950:
5876:
5011:
4413:
4235:
2660:
2583:
2387:
1604:
1379:
474:
238:
Geometric representation of the 2nd to 6th root of a general complex number in polar form. For the
192:
7129:
6774:{\displaystyle \sum _{k=1}^{n}{\overline {z^{j\cdot k}}}\cdot z^{j'\cdot k}=n\cdot \delta _{j,j'}}
4759:
2539:
1541:
10899:
10765:
10676:
9827:
9740:
9156:
7120:
5287:
4674:
4230:
4044:
2990:
2937:
2758:
1406:
1235:
1019:
700:
478:
10575:
10488:
9966:
9253:
8480:
6481:
6073:
3993:
3643:
3068:
9678:
5033:, the four primitive eighth roots of unity are the square roots of the primitive fourth roots,
3637:
1788:
10880:
10871:
10862:
10760:
10735:
10680:
10650:
10628:
10593:
10517:
10382:
10284:
10270:
10246:
10240:
10211:
10205:
10181:
10175:
10154:
10116:
10102:
10077:
10035:
10027:
10006:
9830:
9786:
9725:
9661:
9612:
9497:
4882:
3825:
2941:
2933:
886:
200:
155:
10140:
10071:
10000:
10857:
10852:
10847:
10842:
10837:
10770:
10755:
10724:
10611:
10535:
10466:
10356:
10320:
10276:
10201:
10146:
10108:
9951:
9916:
9878:
9852:
9804:
9763:
9312:
9273:
9257:
9245:
7459:
5462:
3911:
3625:
2877:
of two such automorphisms is obtained by multiplying the exponents. It follows that the map
2654:
2391:
689:
649:{\displaystyle z^{n}=1\quad {\text{and}}\quad z^{m}\neq 1{\text{ for }}m=1,2,3,\ldots ,n-1.}
519:. Conversely, every nonzero element in a finite field is a root of unity in that field. See
208:
10607:
10531:
9710:
5865:{\displaystyle \operatorname {SR} (n)={\begin{cases}1,&n=1\\0,&n>1.\end{cases}}}
4376:
4205:
4161:
176:. Roots of unity are used in many branches of mathematics, and are especially important in
93:
10808:
10740:
10664:
10615:
10603:
10539:
10527:
9640:
9616:
9416:
9383:
9248:; that is, matrices that are invariant under cyclic shifts, a fact that also follows from
9234:
6789:
4060:
3599:
2533:
2529:
181:
3624:
on the right). This geometric fact accounts for the term "cyclotomic" in such phrases as
3007:
The real part of the primitive roots of unity are related to one another as roots of the
10462:
9826:
of each root with its complex conjugate (also a 5th root of unity) is an element of the
8880:
all coefficients of all cyclotomic polynomials are 0, 1, or −1. The first exception is
5592:{\displaystyle z=e^{\frac {2\pi i}{n}}=\cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}}
4481:, the only primitive second (square) root of unity is −1, which is also a non-primitive
676:
In the above formula in terms of exponential and trigonometric functions, the primitive
10442:
10266:
10098:
9774:
9501:
8925:
8916:
th cyclotomic polynomial only has coefficients 0, 1 or −1. Thus the first conceivable
6938:
6923:
4068:
3829:
2093:
1443:
318:
306:
165:
161:
17:
9619:
of a cyclotomic field is an abelian extension of the rationals. It follows that every
4055:, addition, subtraction, multiplication and division if and only if it is possible to
10913:
10794:
10784:
10360:
9940:
9683:
9636:
6641:
4405:
3987:
3591:
2986:
2383:
177:
139:
10325:
10308:
6625:{\displaystyle c_{n}(s)=\sum _{a=1 \atop \gcd(a,n)=1}^{n}e^{2\pi i{\frac {a}{n}}s}.}
10823:
10818:
10775:
10340:
9945:
9848:
9745:
9490:
9230:
9199:
9188:
8884:
8733:
5241:
5096:
4081:
4077:
2836:
2480:
2468:
2395:
696:
664:
527:
516:
204:
10446:
9694:
In the complex plane, the corners of the two squares are the eighth roots of unity
7049:{\displaystyle \sum _{k=1}^{n}{\overline {U_{j,k}}}\cdot U_{k,j'}=\delta _{j,j'},}
4877:
As 7 is not a Fermat prime, the seventh roots of unity are the first that require
2989:, and implies thus that the primitive roots of unity may be expressed in terms of
10669:
4452:
Therefore, the only primitive first root of unity is 1, which is a non-primitive
10304:
10236:
9971:
9241:
5891:
4495:
4052:
3790:
It follows from the discussion in the previous section that this is a primitive
3607:
3080:
2213:
146:
35:
9690:
10505:
10470:
10280:
10150:
10112:
9706:
6188:{\displaystyle \operatorname {R} (n)=\bigcup _{d\,|\,n}\operatorname {P} (d),}
3901:
2173:{\displaystyle \operatorname {R} (n)=\bigcup _{d\,|\,n}\operatorname {P} (d),}
6267:{\displaystyle \operatorname {SR} (n)=\sum _{d\,|\,n}\operatorname {SP} (d).}
1191:{\displaystyle z^{a}=z^{b+kn}=z^{b}z^{kn}=z^{b}(z^{n})^{k}=z^{b}1^{k}=z^{b}.}
9819:
9261:
9172:
5489:
5474:
4878:
4604:{\displaystyle {\frac {-1+i{\sqrt {3}}}{2}},\ {\frac {-1-i{\sqrt {3}}}{2}}.}
4522:
8852:{\displaystyle \Phi _{p}(z)={\frac {z^{p}-1}{z-1}}=\sum _{k=0}^{p-1}z^{k}.}
10210:. Vol. 1 (3rd ed.). American Mathematical Society. p. 129.
4868:{\displaystyle {\frac {1+i{\sqrt {3}}}{2}},\ {\frac {1-i{\sqrt {3}}}{2}}.}
4198:
by the standard manipulation on reciprocal polynomials, and the primitive
4776:
may take the two values 1 and −1 (the same value in the two occurrences).
2476:
270:
4677:, which may be explicitly solved in terms of radicals, giving the roots
473:
However, the defining equation of roots of unity is meaningful over any
9714:
8870:
7474:
4416:, that is, every expression of the roots in terms of radicals involves
2715:
2190:
2007:
169:
10145:. Graduate Texts in Mathematics. Vol. 167. Springer. p. 74.
5086:{\displaystyle \pm {\frac {\sqrt {2}}{2}}\pm i{\frac {\sqrt {2}}{2}}.}
10516:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
1581:
but the converse may be false, as shown by the following example. If
1382:
over a field (in this case the field of complex numbers) has at most
10693:
134:
10309:"On the magnitude of the coefficients of the cyclotomic polynomial"
4881:. There are 6 primitive seventh roots of unity, which are pairwise
203:. For fields with a positive characteristic, the roots belong to a
9689:
9677:
8864:
7371:, which is easier than the general assertion, follows by applying
4999:{\displaystyle {\frac {r}{2}}\pm i{\sqrt {1-{\frac {r^{2}}{4}}}},}
3067:
305:
Unless otherwise specified, the roots of unity may be taken to be
233:
211:, every nonzero element of a finite field is a root of unity. Any
199:
of the field is zero, the roots are complex numbers that are also
133:
4364:{\displaystyle \pm i{\sqrt {1-\left({\frac {r}{2}}\right)^{2}}}.}
3180:{\displaystyle \left(\cos x+i\sin x\right)^{n}=\cos nx+i\sin nx.}
10491:
was born in 1811, died in 1832, but wasn't published until 1846.
7289:{\displaystyle \Phi _{n}(z)=\prod _{k=1}^{\varphi (n)}(z-z_{k})}
4673:, the four primitive fifth roots of unity are the roots of this
3573:{\displaystyle \cos {\frac {2\pi }{n}}+i\sin {\frac {2\pi }{n}}}
2922:{\displaystyle k\mapsto \left(\omega \mapsto \omega ^{k}\right)}
10697:
9844:). For two pairs of non-real 5th roots of unity these sums are
7063:
is simply the complex conjugate. (This fact was first noted by
2434:
th roots of unity form an abelian group under multiplication.
29:
10563:
10550:
10275:. Undergraduate Texts in Mathematics. Springer. p. 160.
10107:. Undergraduate Texts in Mathematics. Springer. p. 149.
6933:
operations. However, it follows from the orthogonality that
6902:{\displaystyle U_{j,k}=n^{-{\frac {1}{2}}}\cdot z^{j\cdot k}}
2997:
Galois group of the real part of the primitive roots of unity
875:
This is also true for negative exponents. In particular, the
226:
is a multiple of the (positive) characteristic of the field.
3963:, which counts (among other things) the number of primitive
3780:{\displaystyle e^{2\pi i{\frac {k}{n}}},\quad 0\leq k<n.}
5858:
2327:
of two roots of unity are also roots of unity. In fact, if
10242:
9260:
is considered (for example, a discretized one-dimensional
2835:
is obtained in this way, and these automorphisms form the
10427:
Inui, Teturo; Tanabe, Yukito; Onodera, Yoshitaka (1996).
10347:(1985). "Solvability by radicals is in polynomial time".
9604:{\displaystyle \mathbb {Q} (\exp(2\pi i/n))/\mathbb {Q} }
9478:{\displaystyle \mathbb {Q} (\exp(2\pi i/n))/\mathbb {Q} }
971:{\displaystyle {\frac {1}{z}}=z^{-1}=z^{n-1}={\bar {z}}.}
7195:
is defined by the fact that its zeros are precisely the
10404:"Solving Cyclotomic Polynomials by Radical Expressions"
9803:, none of the non-real roots of unity (which satisfy a
7090:
algorithms reduces the number of operations further to
5014:, and any such expression involves non-real cube roots.
865:{\displaystyle (z^{k})^{n}=z^{kn}=(z^{n})^{k}=1^{k}=1.}
4494:. With the preceding case, this completes the list of
4404:
is a product of a power of two by a product (possibly
9550:
9510:
9424:
9396:
9320:
9288:
9052:
8993:
8934:
8750:
8293:
7588:
7558:{\displaystyle z^{n}-1=\prod _{d\,|\,n}\Phi _{d}(z).}
7493:
7384:
7211:
7132:
6950:
6840:
6672:
6516:
6290:
6207:
6128:
6014:
5790:
5630:
5508:
5303:
5046:
4949:
4891:
4805:
4762:
4683:
4535:
4379:
4316:
4286:
4238:
4208:
4164:
4108:
3996:
3838:
3728:
3654:
3521:
3358:
3227:
3099:
3017:
2955:
2886:
2845:
2810:
2767:
2727:
2663:
2628:
2591:
2542:
2503:
2258:
2113:
1873:
1791:
1735:
1653:
1607:
1544:
1502:
1451:
1409:
1054:
904:
777:
562:
331:
278:
5883:
th roots of unity being the roots of the polynomial
2416:, the product and the multiplicative inverse of two
9229:. The orthogonality relationship also follows from
4202:th roots of unity may be deduced from the roots of
746:, which is the smallest positive integer such that
60:. Unsourced material may be challenged and removed.
10668:
10002:Field Theory and Its Classical Problems, Volume 14
9603:
9534:
9477:
9407:
9366:
9299:
9094:
9039:{\displaystyle p_{1}<p_{2}<\cdots <p_{t}}
9038:
8979:
8851:
8462:
8269:
7557:
7447:
7288:
7166:
7048:
6901:
6773:
6624:
6370:
6266:
6187:
6050:
5864:
5746:
5591:
5425:
5085:
4998:
4935:
4867:
4768:
4748:
4603:
4392:
4363:
4302:
4272:
4221:
4177:
4133:
4011:
3870:
3779:
3700:
3572:
3490:
3338:
3179:
3052:
2975:
2921:
2862:
2827:
2784:
2746:
2674:
2645:
2608:
2550:
2520:
2302:
2172:
1983:
1812:
1774:
1684:
1639:
1573:
1530:
1482:
1435:
1190:
970:
864:
648:
463:
297:
3419:
3288:
8859:Substituting any positive integer ≥ 2 for
6557:
4280:That is, the real part of the primitive root is
1957:
1927:
1792:
1748:
487:, and this allows considering roots of unity in
9670:on the grounds that Weber completed the proof.
9198:, and in fact these groups comprise all of the
7448:{\displaystyle {\frac {(z+1)^{n}-1}{(z+1)-1}},}
6051:{\displaystyle \operatorname {SP} (n)=\mu (n),}
250: = 0. The principal root is in black.
10005:. Cambridge University Press. pp. 84–86.
9187:th roots of unity form under multiplication a
8483:. So the first few cyclotomic polynomials are
6916:. Computing the inverse transformation using
5116:th root of unity, then the sequence of powers
4629:, the two primitive fourth roots of unity are
4525:) roots of unity, which are the roots of this
2475:originated from the fact that this group is a
309:(including the number 1, and the number −1 if
10709:
10585:Grundlehren der mathematischen Wissenschaften
10381:. Yale University Press. pp. §§359–360.
10313:Bulletin of the American Mathematical Society
9623:th root of unity may be expressed in term of
9367:{\displaystyle \mathbb {Q} (\exp(2\pi i/n)).}
7202:th roots of unity, each with multiplicity 1.
4943:and the primitive seventh roots of unity are
4408:) of distinct Fermat primes, and the regular
4051:th root of unity can be expressed using only
2303:{\displaystyle \sum _{d\,|\,n}\varphi (d)=n.}
699:. For the case of roots of unity in rings of
8:
10583:
10429:Group Theory and Its Applications in Physics
9240:The roots of unity appear as entries of the
9167:th root of unity. This was already shown by
1296:th roots of unity and are all distinct. (If
673:th roots of unity, except 1, are primitive.
138:The 5th roots of unity (blue points) in the
10649:(2nd ed.). New York: Springer-Verlag.
9535:{\displaystyle \mathbb {Z} /n\mathbb {Z} .}
8905:has 1 or 2 odd prime factors (for example,
8901:. More precisely, it can be shown that if
5901:, which is either 1 or 0 according whether
5613:to be expressed as a linear combination of
3598:th roots of unity are at the vertices of a
2873:The rules of exponentiation imply that the
1334:would not be primitive.) This implies that
27:Number that has an integer power equal to 1
10716:
10702:
10694:
9793:th root of unity) is a quadratic integer.
9639:can be written out explicitly in terms of
8876:Note that, contrary to first appearances,
7473:th root of unity for exactly one positive
7075:or its inverse to a given vector requires
6080:
5781:th roots of unity, primitive or not. Then
5099:for the real part of a 17th root of unity.
3896:th roots of unity are, by definition, the
3610:, with one vertex at 1 (see the plots for
2747:{\displaystyle \omega \mapsto \omega ^{k}}
2249:, this demonstrates the classical formula
2022:th root of unity, and therefore there are
1685:{\displaystyle 2\not \equiv 4{\pmod {4}}.}
10324:
10177:Introduction to Applied Algebraic Systems
10134:
10132:
9597:
9596:
9591:
9577:
9552:
9551:
9549:
9525:
9524:
9516:
9512:
9511:
9509:
9471:
9470:
9465:
9451:
9426:
9425:
9423:
9398:
9397:
9395:
9347:
9322:
9321:
9319:
9290:
9289:
9287:
9083:
9070:
9057:
9051:
9030:
9011:
8998:
8992:
8980:{\displaystyle n=p_{1}p_{2}\cdots p_{t},}
8968:
8955:
8945:
8933:
8840:
8824:
8813:
8780:
8773:
8755:
8749:
8740:th roots of unity except 1 are primitive
8441:
8433:
8416:
8400:
8395:
8394:
8390:
8368:
8346:
8330:
8325:
8324:
8320:
8298:
8292:
8248:
8226:
8170:
8141:
8128:
8115:
8102:
8089:
8048:
8019:
7991:
7935:
7906:
7893:
7880:
7839:
7816:
7760:
7731:
7690:
7633:
7597:
7589:
7587:
7537:
7527:
7522:
7521:
7517:
7498:
7492:
7404:
7385:
7383:
7277:
7249:
7238:
7216:
7210:
7152:
7131:
7026:
7002:
6978:
6972:
6966:
6955:
6949:
6887:
6868:
6864:
6845:
6839:
6754:
6724:
6700:
6694:
6688:
6677:
6671:
6604:
6594:
6584:
6543:
6521:
6515:
6351:
6323:
6318:
6317:
6313:
6289:
6240:
6235:
6234:
6230:
6206:
6161:
6156:
6155:
6151:
6127:
6013:
5809:
5789:
5720:
5708:
5698:
5670:
5658:
5648:
5635:
5629:
5574:
5547:
5519:
5507:
5411:
5398:
5373:
5363:
5344:
5331:
5321:
5308:
5302:
5068:
5050:
5045:
4980:
4974:
4966:
4950:
4948:
4909:
4896:
4890:
4849:
4837:
4818:
4806:
4804:
4761:
4730:
4715:
4690:
4684:
4682:
4585:
4570:
4551:
4536:
4534:
4384:
4378:
4350:
4336:
4323:
4315:
4287:
4285:
4243:
4237:
4213:
4207:
4169:
4163:
4121:
4107:
4002:
3997:
3995:
3857:
3837:
3743:
3733:
3727:
3715:, can be used to put the formula for the
3659:
3653:
3555:
3528:
3520:
3464:
3434:
3418:
3398:
3371:
3357:
3287:
3267:
3240:
3226:
3135:
3098:
3036:
3016:
2957:
2956:
2954:
2908:
2885:
2847:
2846:
2844:
2812:
2811:
2809:
2769:
2768:
2766:
2738:
2726:
2665:
2664:
2662:
2630:
2629:
2627:
2593:
2592:
2590:
2544:
2543:
2541:
2505:
2504:
2502:
2471:. It is worth remarking that the term of
2273:
2268:
2267:
2263:
2257:
2146:
2141:
2140:
2136:
2112:
1951:
1913:
1880:
1872:
1790:
1775:{\displaystyle a={\frac {n}{\gcd(k,n)}},}
1742:
1734:
1663:
1652:
1625:
1612:
1606:
1562:
1549:
1543:
1512:
1501:
1461:
1450:
1427:
1414:
1408:
1179:
1166:
1156:
1143:
1133:
1120:
1104:
1094:
1072:
1059:
1053:
954:
953:
938:
922:
905:
903:
850:
837:
827:
808:
795:
785:
776:
602:
590:
580:
567:
561:
406:
376:
342:
330:
283:
277:
120:Learn how and when to remove this message
10231:
10229:
10227:
10180:. Oxford University Press. p. 137.
9653:was published many years before Galois.
9175:exist for calculating such expressions.
2055:is a prime number, all the roots except
1391:From the preceding, it follows that, if
10349:Journal of Computer and System Sciences
9988:
9155:Cyclotomic polynomials are solvable in
8922:3 ⋅ 5 ⋅ 7 = 105
7071:.) The straightforward application of
4095:th root of unity, the same is true for
4057:construct with compass and straightedge
10272:Introduction to Analytic Number Theory
9807:) is a quadratic integer, but the sum
9221:th roots of unity form an irreducible
6640:From the summation formula follows an
5255:-periodic sequence of complex numbers
4037:
3701:{\displaystyle e^{ix}=\cos x+i\sin x,}
2976:{\displaystyle \mathbb {Q} (\omega ).}
2382:Therefore, the roots of unity form an
680:th roots of unity are those for which
9210:for this cyclic group is a primitive
9095:{\displaystyle p_{1}+p_{2}>p_{t},}
5248:-periodic sequences. This means that
4080:or the product of a power of two and
2985:This shows that this Galois group is
2863:{\displaystyle \mathbb {Q} (\omega )}
2828:{\displaystyle \mathbb {Q} (\omega )}
2785:{\displaystyle \mathbb {Q} (\omega )}
2646:{\displaystyle \mathbb {Q} (\omega )}
2609:{\displaystyle \mathbb {Q} (\omega )}
2521:{\displaystyle \mathbb {Q} (\omega )}
1483:{\displaystyle a\equiv b{\pmod {n}}.}
191:Roots of unity can be defined in any
7:
10402:Weber, Andreas; Keckeisen, Michael.
5875:This is an immediate consequence of
5469:is a (discrete) time variable, then
3794:th-root if and only if the fraction
1531:{\displaystyle a\equiv b{\pmod {n}}}
58:adding citations to reliable sources
7363:has integer coefficients and is an
5925:there is nothing to prove, and for
4424:Explicit expressions in low degrees
2461:. This means that the group of the
1671:
1520:
1469:
505:are either complex numbers, if the
265:is a positive integer, is a number
8752:
8295:
7534:
7213:
6544:
6167:
6129:
4134:{\displaystyle r=z+{\frac {1}{z}}}
3832:of the root of unity; that is, as
3816:is in lowest terms; that is, that
2428:th roots of unity. Therefore, the
2152:
2114:
1832:. This results from the fact that
1261:th root of unity. Then the powers
697:Finite field § Roots of unity
549:th root of unity for some smaller
25:
10647:Introduction to Cyclotomic Fields
10059:. Dover Publications. p. 52.
4936:{\displaystyle r^{3}+r^{2}-2r-1,}
3003:Minimal polynomial of 2cos(2pi/n)
2870:over the field of the rationals.
515:is 0, or, otherwise, belong to a
10893:
10076:. World Scientific. p. 36.
9256:. In particular, if a circulant
9206:of the complex number field. A
7469:th root of unity is a primitive
6001:be the sum of all the primitive
4487:th root of unity for every even
4036:th root). (For more details see
3969:th roots of unity. The roots of
3719:th roots of unity into the form
2804:th power. Every automorphism of
2702:th root of unity if and only if
317:, which are complex with a zero
34:
10447:"The discrete cosine transform"
10326:10.1090/S0002-9904-1936-06309-3
10057:Fundamental Concepts of Algebra
9915:equals to either 0, ±1, ±2 or ±
9113:occurs as a coefficient in the
4436:, the cyclotomic polynomial is
4303:{\displaystyle {\frac {r}{2}},}
3758:
3590:This formula shows that in the
3053:{\displaystyle 2\cos(2\pi /n).}
2576:th root of unity is a power of
1664:
1513:
1462:
1018:. Indeed, by the definition of
585:
579:
537:th root of unity is said to be
427:
222:th roots of unity, except when
45:needs additional citations for
10034:. Springer. pp. 276–277.
9957:Group scheme of roots of unity
9674:Relation to quadratic integers
9588:
9585:
9565:
9556:
9462:
9459:
9439:
9430:
9390:th cyclotomic polynomial over
9358:
9355:
9335:
9326:
8767:
8761:
8396:
8378:
8372:
8326:
8310:
8304:
8260:
8241:
8238:
8219:
8216:
8204:
8201:
8189:
8159:
8082:
8079:
8067:
8037:
8012:
8009:
7984:
7981:
7969:
7966:
7954:
7924:
7873:
7870:
7858:
7828:
7809:
7806:
7794:
7791:
7779:
7749:
7724:
7721:
7709:
7679:
7667:
7664:
7652:
7549:
7543:
7523:
7430:
7418:
7401:
7388:
7283:
7264:
7259:
7253:
7228:
7222:
7142:
7136:
6572:
6560:
6533:
6527:
6338:
6332:
6319:
6303:
6297:
6258:
6252:
6236:
6220:
6214:
6179:
6173:
6157:
6141:
6135:
6103:is the set of primitive ones,
6042:
6036:
6027:
6021:
5803:
5797:
4189:as a root may be deduced from
3871:{\displaystyle \cos(2\pi k/n)}
3865:
3845:
3632:; it is from the Greek roots "
3044:
3027:
2967:
2961:
2901:
2890:
2857:
2851:
2822:
2816:
2779:
2773:
2731:
2640:
2634:
2603:
2597:
2515:
2509:
2487:Galois group of the primitive
2288:
2282:
2269:
2189:goes through all the positive
2183:where the notation means that
2164:
2158:
2142:
2126:
2120:
2084:is the set of primitive ones,
1972:
1960:
1942:
1930:
1901:
1889:
1807:
1795:
1763:
1751:
1675:
1665:
1524:
1514:
1473:
1463:
1140:
1126:
959:
834:
820:
792:
778:
1:
10514:Graduate Texts in Mathematics
10070:Moskowitz, Martin A. (2003).
9408:{\displaystyle \mathbb {Q} .}
9382:th roots of unity and is the
9300:{\displaystyle \mathbb {Q} ,}
9225:of any cyclic group of order
7181:th roots of unity, each with
4273:{\displaystyle z^{2}-rz+1=0.}
3828:that can be expressed as the
2675:{\displaystyle \mathbb {Q} .}
1640:{\displaystyle z^{2}=z^{4}=1}
10361:10.1016/0022-0000(85)90013-3
9877:equals to either 0, ±2, or ±
7568:This formula represents the
7167:{\displaystyle p(z)=z^{n}-1}
7067:when solving the problem of
6990:
6712:
6499:, defined as the sum of the
4769:{\displaystyle \varepsilon }
3914:. As this polynomial is not
3711:which is valid for all real
3218:th root of unity – one gets
2551:{\displaystyle \mathbb {Q} }
1838:is the smallest multiple of
1574:{\displaystyle z^{a}=z^{b},}
1369:th roots of unity, since an
762:th root of unity is also an
10804:Quadratic irrational number
10790:Pisot–Vijayaraghavan number
10627:. Berlin: Springer-Verlag.
10379:Disquisitiones Arithmeticae
9646:Disquisitiones Arithmeticae
9250:group representation theory
9233:principles as described in
7069:trigonometric interpolation
6503:th powers of the primitive
6111:is a disjoint union of the
5495:Choosing for the primitive
4458:th root of unity for every
2422:th roots of unity are also
2386:under multiplication. This
2319:Group of all roots of unity
1842:that is also a multiple of
1436:{\displaystyle z^{a}=z^{b}}
10956:
10667:(2006). "Roots of Unity".
10580:Algebraische Zahlentheorie
10174:Reilly, Norman R. (2009).
10055:Meserve, Bruce E. (1982).
9948:, the unit complex numbers
9903:, for any root of unity,
9271:
9117:th cyclotomic polynomial.
7579:into irreducible factors:
7112:
6914:discrete Fourier transform
5759:discrete Fourier transform
5236:} of these sequences is a
4310:and its imaginary part is
4141:is twice the real part of
4012:{\displaystyle {\sqrt{1}}}
3978:are exactly the primitive
3000:
1706:th root of unity. A power
736:th root of unity for some
213:algebraically closed field
186:discrete Fourier transform
10889:
10731:
10623:Neukirch, Jürgen (1986).
10588:. Vol. 322. Berlin:
10551:"Algebraic Number Theory"
10471:10.1137/S0036144598336745
10281:10.1007/978-1-4757-5579-4
10245:. Springer. p. 306.
10151:10.1007/978-1-4612-4040-2
10139:Morandi, Patrick (1996).
10113:10.1007/978-1-4615-6465-2
9865:, for any root of unity
9781:For four other values of
9724:, the roots of unity are
9278:By adjoining a primitive
6468:This is the special case
5436:for some complex numbers
5290:of powers of a primitive
3079:, which is valid for all
2039:th roots of unity (where
1813:{\displaystyle \gcd(k,n)}
1328:, which would imply that
1201:Therefore, given a power
493:. Whichever is the field
321:), and in this case, the
10562:Milne, James S. (1997).
10104:Applied Abstract Algebra
10073:Adventure in Mathematics
8897:prime factors appear in
7347:Euler's totient function
7059:and thus the inverse of
6279:Möbius inversion formula
6005:th roots of unity. Then
5219:). Furthermore, the set
4084:that are all different.
4038:§ Cyclotomic fields
3961:Euler's totient function
3064:Trigonometric expression
2949:and the Galois group of
2798:th root of unity to its
2718:. In this case, the map
2049:). This implies that if
2047:Euler's totient function
1234:is the remainder of the
885:th root of unity is its
756:Any integer power of an
499:, the roots of unity in
298:{\displaystyle z^{n}=1.}
153:, occasionally called a
10643:Washington, Lawrence C.
10487:was published in 1801,
10207:Advanced Modern Algebra
10142:Field and Galois theory
9667:Kronecker–Weber theorem
9643:: this theory from the
9611:is abelian, this is an
9544:As the Galois group of
7337:th roots of unity, and
6083:, it was shown that if
5967:, so the sum satisfies
5949:th roots of unity is a
4521:, the primitive third (
3072:The cube roots of unity
2622:th roots of unity, and
2467:th roots of unity is a
2455:th roots are powers of
1822:greatest common divisor
1496:is not primitive then
1403:th root of unity, then
18:Primitive root of unity
10900:Mathematics portal
10584:
9705:, both roots of unity
9695:
9687:
9605:
9536:
9500:to the multiplicative
9479:
9409:
9368:
9301:
9096:
9040:
8981:
8853:
8835:
8464:
8271:
7559:
7458:and expanding via the
7449:
7373:Eisenstein's criterion
7365:irreducible polynomial
7290:
7263:
7168:
7109:Cyclotomic polynomials
7088:fast Fourier transform
7050:
6971:
6903:
6775:
6693:
6626:
6589:
6372:
6268:
6189:
6095:th roots of unity and
6052:
5866:
5777:be the sum of all the
5748:
5593:
5427:
5286:can be expressed as a
5087:
5000:
4937:
4869:
4770:
4750:
4605:
4394:
4365:
4304:
4274:
4223:
4179:
4135:
4032:possible values for a
4013:
3872:
3781:
3702:
3574:
3492:
3340:
3181:
3073:
3054:
2977:
2923:
2864:
2829:
2786:
2748:
2676:
2647:
2610:
2552:
2522:
2325:multiplicative inverse
2304:
2174:
2076:th roots of unity and
1985:
1814:
1776:
1726:th root of unity for
1686:
1641:
1575:
1532:
1484:
1437:
1192:
972:
866:
650:
465:
325:th roots of unity are
299:
251:
242:th root of unity, set
142:
9977:Teichmüller character
9693:
9681:
9664:, usually called the
9627:-roots, with various
9606:
9537:
9480:
9410:
9369:
9302:
9097:
9041:
8982:
8863:, this sum becomes a
8854:
8809:
8744:th roots. Therefore,
8465:
8284:to the formula gives
8272:
7560:
7450:
7291:
7234:
7192:cyclotomic polynomial
7169:
7115:Cyclotomic polynomial
7051:
6951:
6904:
6776:
6673:
6659: = 1, … ,
6649: = 1, … ,
6627:
6539:
6373:
6269:
6190:
6081:Elementary properties
6053:
5867:
5749:
5594:
5428:
5088:
5001:
4938:
4870:
4771:
4751:
4606:
4395:
4393:{\displaystyle R_{n}}
4366:
4305:
4275:
4224:
4222:{\displaystyle R_{n}}
4180:
4178:{\displaystyle R_{n}}
4156:reciprocal polynomial
4136:
4014:
3939:cyclotomic polynomial
3873:
3782:
3703:
3630:cyclotomic polynomial
3575:
3493:
3341:
3182:
3071:
3055:
2978:
2924:
2865:
2830:
2787:
2749:
2677:
2648:
2611:
2553:
2523:
2365:least common multiple
2305:
2175:
1986:
1854:least common multiple
1815:
1777:
1687:
1642:
1576:
1533:
1485:
1438:
1193:
993:th root of unity and
973:
867:
768:th root of unity, as
714:Elementary properties
705:Root of unity modulo
651:
530:for further details.
521:Root of unity modulo
466:
300:
237:
137:
10746:Constructible number
10675:. Washington, D.C.:
10564:"Class Field Theory"
10026:Lang, Serge (2002).
9720:For three values of
9548:
9508:
9422:
9394:
9318:
9286:
9282:th root of unity to
9204:multiplicative group
9050:
8991:
8932:
8928:. In particular, if
8748:
8291:
7586:
7491:
7484:. This implies that
7382:
7209:
7130:
6948:
6918:Gaussian elimination
6838:
6670:
6514:
6381:In this formula, if
6288:
6205:
6126:
6012:
5932:there exists a root
5788:
5628:
5506:
5301:
5167:sequences of powers
5044:
4947:
4889:
4803:
4760:
4681:
4533:
4527:quadratic polynomial
4377:
4314:
4284:
4236:
4206:
4162:
4106:
3994:
3941:, and often denoted
3886:Algebraic expression
3880:trigonometric number
3836:
3726:
3652:
3519:
3356:
3225:
3097:
3015:
2953:
2884:
2843:
2808:
2765:
2725:
2661:
2626:
2589:
2540:
2501:
2323:The product and the
2256:
2111:
2016:is also a primitive
1871:
1789:
1733:
1651:
1605:
1594:th root of unity is
1542:
1500:
1449:
1407:
1052:
902:
775:
560:
329:
276:
54:improve this article
10872:Supersilver ratio (
10863:Supergolden ratio (
10463:1999SIAMR..41..135S
9997:Hadlock, Charles R.
9962:Dirichlet character
9741:Eisenstein integers
9171:in 1797. Efficient
6507:th roots of unity:
5918:Alternatively, for
5890:, their sum is the
5209:-periodic (because
5142:-periodic (because
5012:casus irreducibilis
4414:casus irreducibilis
4067:. This is the case
3984:th roots of unity.
3512:. In other words,
3077:De Moivre's formula
2792:, which maps every
2062:In other words, if
2033:distinct primitive
1380:polynomial equation
477:(and even over any
246: = 1 and
164:that yields 1 when
10766:Eisenstein integer
10677:Joseph Henry Press
10625:Class Field Theory
9726:quadratic integers
9696:
9688:
9635:). In these cases
9601:
9532:
9475:
9405:
9364:
9297:
9214:th root of unity.
9092:
9036:
8977:
8849:
8460:
8405:
8335:
8267:
8265:
7572:of the polynomial
7555:
7532:
7445:
7375:to the polynomial
7333:are the primitive
7286:
7177:are precisely the
7164:
7123:of the polynomial
7046:
6899:
6800:th root of unity.
6771:
6644:relationship: for
6622:
6368:
6328:
6264:
6245:
6185:
6166:
6091:is the set of all
6048:
5862:
5857:
5744:
5703:
5653:
5589:
5461:This is a form of
5454:and every integer
5423:
5326:
5294:th root of unity:
5288:linear combination
5159:for all values of
5083:
4996:
4933:
4865:
4766:
4746:
4675:quartic polynomial
4601:
4390:
4361:
4300:
4270:
4231:quadratic equation
4219:
4175:
4145:. In other words,
4131:
4009:
3868:
3777:
3698:
3587:th root of unity.
3570:
3488:
3336:
3214:gives a primitive
3177:
3074:
3050:
3009:minimal polynomial
2973:
2919:
2860:
2825:
2782:
2744:
2672:
2643:
2606:
2548:
2518:
2437:Given a primitive
2300:
2278:
2170:
2151:
2070:is the set of all
1981:
1846:. In other words,
1810:
1772:
1682:
1637:
1588:, a non-primitive
1571:
1528:
1480:
1433:
1236:Euclidean division
1188:
1020:congruence modulo
968:
895:th root of unity:
862:
646:
461:
295:
252:
230:General definition
201:algebraic integers
143:
10925:Cyclotomic fields
10920:Algebraic numbers
10907:
10906:
10881:Twelfth root of 2
10761:Doubling the cube
10751:Conway's constant
10736:Algebraic integer
10725:Algebraic numbers
10599:978-3-540-65399-8
10523:978-0-387-95385-4
10290:978-1-4419-2805-4
10202:Rotman, Joseph J.
10187:978-0-19-536787-4
10160:978-0-387-94753-2
10122:978-0-387-96166-8
10041:978-0-387-95385-4
10012:978-0-88385-032-9
9787:complex conjugate
9764:Gaussian integers
9613:abelian extension
9268:Cyclotomic fields
9202:subgroups of the
8910: = 150
8804:
8449:
8386:
8354:
8316:
7513:
7440:
7349:. The polynomial
6993:
6876:
6796:is any primitive
6715:
6612:
6582:
6359:
6309:
6226:
6147:
5739:
5694:
5689:
5644:
5587:
5560:
5535:
5499:th root of unity
5317:
5078:
5074:
5060:
5056:
4991:
4989:
4958:
4883:complex conjugate
4860:
4854:
4836:
4829:
4823:
4741:
4737:
4735:
4707:
4695:
4596:
4590:
4569:
4562:
4556:
4356:
4344:
4295:
4158:, the polynomial
4129:
4047:that a primitive
4007:
3925:), the primitive
3912:algebraic numbers
3826:irrational number
3824:are coprime. An
3751:
3640:" (cut, divide).
3636:" (circle) plus "
3606:inscribed in the
3568:
3541:
3480:
3450:
3411:
3384:
3280:
3253:
2934:group isomorphism
2564:th root of unity
2493:th roots of unity
2443:th root of unity
2408:th roots of unity
2259:
2132:
1976:
1946:
1908:
1767:
1039:for some integer
962:
913:
889:, and is also an
887:complex conjugate
724:th root of unity
605:
583:
422:
392:
361:
215:contains exactly
168:to some positive
130:
129:
122:
104:
16:(Redirected from
10947:
10898:
10897:
10875:
10866:
10858:Square root of 7
10853:Square root of 6
10848:Square root of 5
10843:Square root of 3
10838:Square root of 2
10831:
10827:
10798:
10779:
10771:Gaussian integer
10756:Cyclotomic field
10718:
10711:
10704:
10695:
10690:
10674:
10671:Unknown Quantity
10665:Derbyshire, John
10660:
10638:
10619:
10587:
10576:Neukirch, Jürgen
10571:
10558:
10542:
10492:
10481:
10475:
10474:
10439:
10433:
10432:
10424:
10418:
10417:
10415:
10413:
10408:
10399:
10393:
10392:
10371:
10365:
10364:
10337:
10331:
10330:
10328:
10301:
10295:
10294:
10263:
10257:
10256:
10233:
10222:
10221:
10198:
10192:
10191:
10171:
10165:
10164:
10136:
10127:
10126:
10094:
10088:
10087:
10067:
10061:
10060:
10052:
10046:
10045:
10028:"Roots of unity"
10023:
10017:
10016:
9993:
9952:Cyclotomic field
9930:
9922:
9921:
9914:
9912:
9902:
9892:
9884:
9883:
9876:
9874:
9864:
9843:
9835:
9825:
9816:
9805:quartic equation
9802:
9792:
9784:
9772:
9761:
9751:
9738:
9723:
9704:
9641:Gaussian periods
9631:not exceeding φ(
9610:
9608:
9607:
9602:
9600:
9595:
9581:
9555:
9541:
9539:
9538:
9533:
9528:
9520:
9515:
9484:
9482:
9481:
9476:
9474:
9469:
9455:
9429:
9414:
9412:
9411:
9406:
9401:
9389:
9381:
9373:
9371:
9370:
9365:
9351:
9325:
9313:cyclotomic field
9310:
9307:one obtains the
9306:
9304:
9303:
9298:
9293:
9281:
9274:Cyclotomic field
9258:Hermitian matrix
9252:as a variant of
9246:circulant matrix
9228:
9220:
9213:
9197:
9186:
9166:
9162:
9151:
9140:
9129: ∣ Φ
9123:
9116:
9112:
9101:
9099:
9098:
9093:
9088:
9087:
9075:
9074:
9062:
9061:
9046:are odd primes,
9045:
9043:
9042:
9037:
9035:
9034:
9016:
9015:
9003:
9002:
8986:
8984:
8983:
8978:
8973:
8972:
8960:
8959:
8950:
8949:
8923:
8919:
8915:
8911:
8904:
8900:
8892:
8888:
8868:
8862:
8858:
8856:
8855:
8850:
8845:
8844:
8834:
8823:
8805:
8803:
8792:
8785:
8784:
8774:
8760:
8759:
8743:
8739:
8731:
8724:
8699:
8654:
8617:
8580:
8555:
8526:
8501:
8478:
8469:
8467:
8466:
8461:
8456:
8455:
8454:
8450:
8442:
8432:
8428:
8421:
8420:
8404:
8399:
8382:
8381:
8367:
8363:
8356:
8355:
8347:
8334:
8329:
8303:
8302:
8282:Möbius inversion
8276:
8274:
8273:
8268:
8266:
8253:
8252:
8231:
8230:
8175:
8174:
8146:
8145:
8133:
8132:
8120:
8119:
8107:
8106:
8094:
8093:
8053:
8052:
8024:
8023:
7996:
7995:
7940:
7939:
7911:
7910:
7898:
7897:
7885:
7884:
7844:
7843:
7821:
7820:
7765:
7764:
7736:
7735:
7695:
7694:
7638:
7637:
7602:
7601:
7578:
7564:
7562:
7561:
7556:
7542:
7541:
7531:
7526:
7503:
7502:
7483:
7479:
7472:
7468:
7460:binomial theorem
7454:
7452:
7451:
7446:
7441:
7439:
7416:
7409:
7408:
7386:
7370:
7362:
7344:
7336:
7332:
7295:
7293:
7292:
7287:
7282:
7281:
7262:
7248:
7221:
7220:
7201:
7188:
7180:
7173:
7171:
7170:
7165:
7157:
7156:
7104:
7086:operations. The
7085:
7074:
7062:
7055:
7053:
7052:
7047:
7042:
7041:
7040:
7018:
7017:
7016:
6994:
6989:
6988:
6973:
6970:
6965:
6936:
6932:
6908:
6906:
6905:
6900:
6898:
6897:
6879:
6878:
6877:
6869:
6856:
6855:
6830:
6818:
6812:
6799:
6795:
6787:
6780:
6778:
6777:
6772:
6770:
6769:
6768:
6740:
6739:
6732:
6716:
6711:
6710:
6695:
6692:
6687:
6663:
6653:
6631:
6629:
6628:
6623:
6618:
6617:
6613:
6605:
6588:
6583:
6581:
6555:
6526:
6525:
6506:
6502:
6498:
6479:
6464:
6448:
6446:
6444:
6443:
6438:
6435:
6424:
6414:
6412:
6410:
6409:
6404:
6401:
6390:
6377:
6375:
6374:
6369:
6364:
6360:
6352:
6327:
6322:
6273:
6271:
6270:
6265:
6244:
6239:
6194:
6192:
6191:
6186:
6165:
6160:
6118:
6110:
6102:
6094:
6090:
6071:
6057:
6055:
6054:
6049:
6004:
6000:
5989:
5981:
5966:
5959:
5948:
5944:
5939:– since the set
5938:
5931:
5924:
5914:
5907:
5900:
5889:
5882:
5877:Vieta's formulas
5871:
5869:
5868:
5863:
5861:
5860:
5780:
5776:
5753:
5751:
5750:
5745:
5740:
5735:
5721:
5713:
5712:
5702:
5690:
5685:
5671:
5663:
5662:
5652:
5640:
5639:
5620:
5616:
5612:
5598:
5596:
5595:
5590:
5588:
5583:
5575:
5561:
5556:
5548:
5537:
5536:
5531:
5520:
5498:
5487:
5472:
5468:
5463:Fourier analysis
5457:
5453:
5432:
5430:
5429:
5424:
5422:
5421:
5403:
5402:
5384:
5383:
5368:
5367:
5355:
5354:
5336:
5335:
5325:
5313:
5312:
5293:
5282:
5254:
5247:
5235:
5218:
5208:
5204:
5191:
5166:
5162:
5158:
5141:
5134:
5115:
5111:
5092:
5090:
5089:
5084:
5079:
5070:
5069:
5061:
5052:
5051:
5040:. They are thus
5039:
5032:
5009:
5005:
5003:
5002:
4997:
4992:
4990:
4985:
4984:
4975:
4967:
4959:
4951:
4942:
4940:
4939:
4934:
4914:
4913:
4901:
4900:
4874:
4872:
4871:
4866:
4861:
4856:
4855:
4850:
4838:
4834:
4830:
4825:
4824:
4819:
4807:
4798:
4775:
4773:
4772:
4767:
4755:
4753:
4752:
4747:
4742:
4736:
4731:
4717:
4716:
4708:
4703:
4696:
4691:
4685:
4672:
4641:
4634:
4628:
4610:
4608:
4607:
4602:
4597:
4592:
4591:
4586:
4571:
4567:
4563:
4558:
4557:
4552:
4537:
4520:
4493:
4486:
4480:
4457:
4451:
4435:
4418:nonreal radicals
4411:
4403:
4399:
4397:
4396:
4391:
4389:
4388:
4370:
4368:
4367:
4362:
4357:
4355:
4354:
4349:
4345:
4337:
4324:
4309:
4307:
4306:
4301:
4296:
4288:
4279:
4277:
4276:
4271:
4248:
4247:
4228:
4226:
4225:
4220:
4218:
4217:
4201:
4197:
4188:
4184:
4182:
4181:
4176:
4174:
4173:
4153:
4144:
4140:
4138:
4137:
4132:
4130:
4122:
4101:
4094:
4090:
4075:
4064:
4050:
4035:
4031:
4027:
4023:
4018:
4016:
4015:
4010:
4008:
4006:
3998:
3983:
3977:
3968:
3958:
3950:. The degree of
3949:
3936:
3930:
3924:
3909:
3895:
3877:
3875:
3874:
3869:
3861:
3823:
3819:
3815:
3814:
3812:
3811:
3806:
3803:
3793:
3786:
3784:
3783:
3778:
3754:
3753:
3752:
3744:
3718:
3714:
3707:
3705:
3704:
3699:
3667:
3666:
3626:cyclotomic field
3623:
3616:
3603:
3597:
3586:
3579:
3577:
3576:
3571:
3569:
3564:
3556:
3542:
3537:
3529:
3511:
3497:
3495:
3494:
3489:
3481:
3476:
3465:
3451:
3446:
3435:
3424:
3423:
3417:
3413:
3412:
3407:
3399:
3385:
3380:
3372:
3345:
3343:
3342:
3337:
3293:
3292:
3286:
3282:
3281:
3276:
3268:
3254:
3249:
3241:
3217:
3213:
3212:
3210:
3209:
3204:
3201:
3186:
3184:
3183:
3178:
3140:
3139:
3134:
3130:
3089:
3085:
3059:
3057:
3056:
3051:
3040:
2982:
2980:
2979:
2974:
2960:
2947:
2942:integers modulo
2928:
2926:
2925:
2920:
2918:
2914:
2913:
2912:
2869:
2867:
2866:
2861:
2850:
2834:
2832:
2831:
2826:
2815:
2803:
2797:
2791:
2789:
2788:
2783:
2772:
2753:
2751:
2750:
2745:
2743:
2742:
2713:
2707:
2701:
2695:
2689:
2681:
2679:
2678:
2673:
2668:
2655:Galois extension
2652:
2650:
2649:
2644:
2633:
2621:
2615:
2613:
2612:
2607:
2596:
2581:
2575:
2569:
2563:
2557:
2555:
2554:
2549:
2547:
2534:rational numbers
2527:
2525:
2524:
2519:
2508:
2492:
2466:
2460:
2454:
2448:
2442:
2433:
2427:
2421:
2407:
2392:torsion subgroup
2378:
2372:
2362:
2356:
2348:
2340:
2333:
2314:Group properties
2309:
2307:
2306:
2301:
2277:
2272:
2248:
2237:
2229:
2223:
2208:
2202:
2198:
2188:
2179:
2177:
2176:
2171:
2150:
2145:
2103:
2091:
2083:
2075:
2069:
2058:
2054:
2044:
2038:
2032:
2021:
2015:
2005:
1999:
1990:
1988:
1987:
1982:
1977:
1975:
1952:
1947:
1945:
1922:
1914:
1909:
1904:
1881:
1863:
1859:
1851:
1845:
1841:
1837:
1831:
1827:
1819:
1817:
1816:
1811:
1781:
1779:
1778:
1773:
1768:
1766:
1743:
1725:
1719:
1715:
1705:
1699:
1691:
1689:
1688:
1683:
1678:
1646:
1644:
1643:
1638:
1630:
1629:
1617:
1616:
1600:
1593:
1587:
1580:
1578:
1577:
1572:
1567:
1566:
1554:
1553:
1537:
1535:
1534:
1529:
1527:
1495:
1489:
1487:
1486:
1481:
1476:
1442:
1440:
1439:
1434:
1432:
1431:
1419:
1418:
1402:
1396:
1387:
1374:
1368:
1362:
1351:
1345:
1339:
1333:
1327:
1320:
1305:
1295:
1289:
1278:
1272:
1266:
1260:
1254:
1245:
1241:
1233:
1222:
1212:
1206:
1197:
1195:
1194:
1189:
1184:
1183:
1171:
1170:
1161:
1160:
1148:
1147:
1138:
1137:
1125:
1124:
1112:
1111:
1099:
1098:
1086:
1085:
1064:
1063:
1044:
1038:
1017:
1007:
992:
986:
977:
975:
974:
969:
964:
963:
955:
949:
948:
930:
929:
914:
906:
894:
884:
871:
869:
868:
863:
855:
854:
842:
841:
832:
831:
816:
815:
800:
799:
790:
789:
767:
761:
752:
745:
735:
729:
723:
701:modular integers
690:coprime integers
687:
683:
679:
672:
655:
653:
652:
647:
606:
603:
595:
594:
584:
581:
572:
571:
552:
548:
545:if it is not an
543:
542:
536:
514:
504:
498:
492:
486:
470:
468:
467:
462:
423:
418:
407:
393:
388:
377:
366:
362:
357:
343:
324:
312:
304:
302:
301:
296:
288:
287:
268:
264:
259:th root of unity
258:
249:
245:
225:
221:
218:
182:group characters
180:, the theory of
175:
125:
118:
114:
111:
105:
103:
62:
38:
30:
21:
10955:
10954:
10950:
10949:
10948:
10946:
10945:
10944:
10940:Complex numbers
10910:
10909:
10908:
10903:
10892:
10885:
10873:
10864:
10832:
10829:
10825:
10809:Rational number
10796:
10795:Plastic ratio (
10777:
10741:Chebyshev nodes
10727:
10722:
10687:
10663:
10657:
10641:
10635:
10622:
10600:
10590:Springer-Verlag
10574:
10561:
10547:Milne, James S.
10545:
10524:
10504:
10501:
10496:
10495:
10482:
10478:
10443:Strang, Gilbert
10441:
10440:
10436:
10426:
10425:
10421:
10411:
10409:
10406:
10401:
10400:
10396:
10389:
10373:
10372:
10368:
10345:Miller, Gary L.
10339:
10338:
10334:
10303:
10302:
10298:
10291:
10267:Apostol, Tom M.
10265:
10264:
10260:
10253:
10235:
10234:
10225:
10218:
10200:
10199:
10195:
10188:
10173:
10172:
10168:
10161:
10138:
10137:
10130:
10123:
10096:
10095:
10091:
10084:
10069:
10068:
10064:
10054:
10053:
10049:
10042:
10025:
10024:
10020:
10013:
9995:
9994:
9990:
9985:
9967:Ramanujan's sum
9937:
9925:
9919:
9917:
9910:
9904:
9897:
9887:
9881:
9879:
9872:
9866:
9859:
9838:
9831:
9814:
9808:
9797:
9790:
9782:
9767:
9756:
9744:
9733:
9721:
9699:
9676:
9546:
9545:
9506:
9505:
9420:
9419:
9417:field extension
9392:
9391:
9387:
9384:splitting field
9379:
9316:
9315:
9308:
9284:
9283:
9279:
9276:
9270:
9254:Bloch's theorem
9235:Character group
9231:group-theoretic
9226:
9218:
9211:
9195:
9184:
9181:
9164:
9160:
9142:
9141:if and only if
9134:
9125:
9124:is prime, then
9121:
9114:
9107:
9079:
9066:
9053:
9048:
9047:
9026:
9007:
8994:
8989:
8988:
8964:
8951:
8941:
8930:
8929:
8921:
8917:
8913:
8906:
8902:
8898:
8893:as on how many
8890:
8887:
8881:
8866:
8860:
8836:
8793:
8776:
8775:
8751:
8746:
8745:
8741:
8737:
8736:, then all the
8729:
8706:
8702:
8661:
8657:
8624:
8620:
8587:
8583:
8562:
8558:
8533:
8529:
8508:
8504:
8491:
8487:
8481:Möbius function
8474:
8437:
8412:
8411:
8407:
8406:
8342:
8341:
8337:
8336:
8294:
8289:
8288:
8264:
8263:
8244:
8222:
8182:
8166:
8163:
8162:
8137:
8124:
8111:
8098:
8085:
8060:
8044:
8041:
8040:
8015:
7987:
7947:
7931:
7928:
7927:
7902:
7889:
7876:
7851:
7835:
7832:
7831:
7812:
7772:
7756:
7753:
7752:
7727:
7702:
7686:
7683:
7682:
7645:
7629:
7626:
7625:
7609:
7593:
7584:
7583:
7573:
7533:
7494:
7489:
7488:
7481:
7477:
7470:
7466:
7417:
7400:
7387:
7380:
7379:
7368:
7356:
7350:
7338:
7334:
7331:
7320:
7313:
7306:
7300:
7273:
7212:
7207:
7206:
7199:
7186:
7178:
7148:
7128:
7127:
7117:
7111:
7091:
7076:
7072:
7060:
7033:
7022:
7009:
6998:
6974:
6946:
6945:
6934:
6921:
6883:
6860:
6841:
6836:
6835:
6820:
6816:
6808: ×
6804:
6797:
6793:
6790:Kronecker delta
6785:
6761:
6750:
6725:
6720:
6696:
6668:
6667:
6655:
6645:
6638:
6590:
6556:
6545:
6517:
6512:
6511:
6504:
6500:
6492:
6484:
6482:Ramanujan's sum
6477:
6469:
6450:
6439:
6436:
6431:
6430:
6428:
6426:
6416:
6405:
6402:
6397:
6396:
6394:
6392:
6382:
6347:
6286:
6285:
6203:
6202:
6124:
6123:
6112:
6104:
6096:
6092:
6084:
6079:In the section
6074:Möbius function
6062:
6010:
6009:
6002:
5994:
5983:
5968:
5957:
5954:
5946:
5940:
5933:
5926:
5919:
5909:
5902:
5895:
5884:
5880:
5879:. In fact, the
5856:
5855:
5844:
5835:
5834:
5823:
5810:
5786:
5785:
5778:
5770:
5767:
5722:
5704:
5672:
5654:
5631:
5626:
5625:
5618:
5614:
5611:
5603:
5576:
5549:
5521:
5515:
5504:
5503:
5496:
5486:
5478:
5470:
5466:
5455:
5452:
5443:
5437:
5407:
5394:
5369:
5359:
5340:
5327:
5304:
5299:
5298:
5291:
5280:
5273:
5266:
5259:
5252:
5245:
5233:
5227:
5220:
5210:
5206:
5196:
5176:
5171:
5164:
5160:
5143:
5139:
5120:
5113:
5112:is a primitive
5109:
5106:
5042:
5041:
5034:
5022:
5018:
5007:
4976:
4945:
4944:
4905:
4892:
4887:
4886:
4839:
4808:
4801:
4800:
4784:
4780:
4758:
4757:
4686:
4679:
4678:
4650:
4646:
4636:
4630:
4618:
4614:
4572:
4538:
4531:
4530:
4506:
4502:
4498:roots of unity.
4488:
4482:
4470:
4466:
4453:
4441:
4437:
4430:
4426:
4409:
4401:
4380:
4375:
4374:
4373:The polynomial
4332:
4331:
4312:
4311:
4282:
4281:
4239:
4234:
4233:
4229:by solving the
4209:
4204:
4203:
4199:
4196:
4190:
4186:
4165:
4160:
4159:
4152:
4146:
4142:
4104:
4103:
4096:
4092:
4091:is a primitive
4088:
4071:
4062:
4048:
4033:
4029:
4025:
4021:
3992:
3991:
3979:
3976:
3970:
3964:
3957:
3951:
3948:
3942:
3932:
3926:
3919:
3910:, and are thus
3904:
3891:
3888:
3834:
3833:
3821:
3817:
3807:
3804:
3799:
3798:
3796:
3795:
3791:
3729:
3724:
3723:
3716:
3712:
3655:
3650:
3649:
3644:Euler's formula
3618:
3611:
3601:
3595:
3584:
3583:is a primitive
3557:
3530:
3517:
3516:
3502:
3466:
3436:
3400:
3373:
3364:
3360:
3359:
3354:
3353:
3269:
3242:
3233:
3229:
3228:
3223:
3222:
3215:
3205:
3202:
3199:
3198:
3196:
3191:
3105:
3101:
3100:
3095:
3094:
3087:
3083:
3066:
3013:
3012:
3005:
2999:
2951:
2950:
2943:
2940:of the ring of
2904:
2897:
2893:
2882:
2881:
2841:
2840:
2806:
2805:
2799:
2793:
2763:
2762:
2734:
2723:
2722:
2709:
2703:
2697:
2696:is a primitive
2691:
2690:is an integer,
2685:
2659:
2658:
2624:
2623:
2617:
2587:
2586:
2577:
2571:
2565:
2559:
2558:by a primitive
2538:
2537:
2536:generated over
2530:field extension
2499:
2498:
2495:
2488:
2462:
2456:
2450:
2444:
2438:
2429:
2423:
2417:
2412:For an integer
2410:
2403:
2374:
2368:
2358:
2350:
2342:
2335:
2328:
2321:
2316:
2254:
2253:
2239:
2231:
2225:
2217:
2204:
2200:
2194:
2184:
2109:
2108:
2097:
2085:
2077:
2071:
2063:
2059:are primitive.
2056:
2050:
2040:
2034:
2023:
2017:
2011:
2001:
1995:
1956:
1923:
1915:
1882:
1869:
1868:
1861:
1857:
1847:
1843:
1839:
1833:
1829:
1825:
1787:
1786:
1747:
1731:
1730:
1721:
1720:is a primitive
1717:
1707:
1701:
1700:be a primitive
1695:
1649:
1648:
1621:
1608:
1603:
1602:
1595:
1589:
1582:
1558:
1545:
1540:
1539:
1498:
1497:
1491:
1447:
1446:
1423:
1410:
1405:
1404:
1398:
1397:is a primitive
1392:
1383:
1370:
1364:
1363:are all of the
1353:
1347:
1341:
1335:
1329:
1322:
1307:
1297:
1291:
1280:
1274:
1268:
1262:
1256:
1255:be a primitive
1250:
1243:
1239:
1224:
1214:
1208:
1202:
1175:
1162:
1152:
1139:
1129:
1116:
1100:
1090:
1068:
1055:
1050:
1049:
1040:
1026:
1009:
994:
988:
982:
934:
918:
900:
899:
890:
880:
846:
833:
823:
804:
791:
781:
773:
772:
763:
757:
747:
737:
731:
730:is a primitive
725:
719:
716:
685:
681:
677:
668:
604: for
586:
563:
558:
557:
550:
546:
540:
539:
534:
510:
500:
494:
488:
482:
408:
378:
344:
338:
327:
326:
322:
310:
307:complex numbers
279:
274:
273:
269:satisfying the
266:
262:
256:
247:
243:
232:
223:
219:
216:
173:
126:
115:
109:
106:
69:"Root of unity"
63:
61:
51:
39:
28:
23:
22:
15:
12:
11:
5:
10953:
10951:
10943:
10942:
10937:
10932:
10927:
10922:
10912:
10911:
10905:
10904:
10890:
10887:
10886:
10884:
10883:
10878:
10869:
10860:
10855:
10850:
10845:
10840:
10835:
10828:
10824:Silver ratio (
10821:
10816:
10811:
10806:
10801:
10792:
10787:
10782:
10776:Golden ratio (
10773:
10768:
10763:
10758:
10753:
10748:
10743:
10738:
10732:
10729:
10728:
10723:
10721:
10720:
10713:
10706:
10698:
10692:
10691:
10685:
10661:
10655:
10639:
10633:
10620:
10598:
10572:
10559:
10543:
10522:
10500:
10497:
10494:
10493:
10485:Disquisitiones
10476:
10457:(1): 135–147.
10434:
10419:
10394:
10387:
10375:Gauss, Carl F.
10366:
10355:(2): 179–208.
10332:
10319:(6): 389–392.
10296:
10289:
10258:
10251:
10223:
10216:
10193:
10186:
10166:
10159:
10128:
10121:
10097:Lidl, Rudolf;
10089:
10082:
10062:
10047:
10040:
10018:
10011:
9987:
9986:
9984:
9981:
9980:
9979:
9974:
9969:
9964:
9959:
9954:
9949:
9943:
9936:
9933:
9855:golden ratio.
9779:
9778:
9775:Imaginary unit
9753:
9675:
9672:
9599:
9594:
9590:
9587:
9584:
9580:
9576:
9573:
9570:
9567:
9564:
9561:
9558:
9554:
9531:
9527:
9523:
9519:
9514:
9502:group of units
9473:
9468:
9464:
9461:
9458:
9454:
9450:
9447:
9444:
9441:
9438:
9435:
9432:
9428:
9404:
9400:
9363:
9360:
9357:
9354:
9350:
9346:
9343:
9340:
9337:
9334:
9331:
9328:
9324:
9296:
9292:
9272:Main article:
9269:
9266:
9223:representation
9180:
9177:
9130:
9091:
9086:
9082:
9078:
9073:
9069:
9065:
9060:
9056:
9033:
9029:
9025:
9022:
9019:
9014:
9010:
9006:
9001:
8997:
8976:
8971:
8967:
8963:
8958:
8954:
8948:
8944:
8940:
8937:
8926:absolute value
8883:
8848:
8843:
8839:
8833:
8830:
8827:
8822:
8819:
8816:
8812:
8808:
8802:
8799:
8796:
8791:
8788:
8783:
8779:
8772:
8769:
8766:
8763:
8758:
8754:
8726:
8725:
8704:
8700:
8659:
8655:
8622:
8618:
8585:
8581:
8560:
8556:
8531:
8527:
8506:
8502:
8489:
8471:
8470:
8459:
8453:
8448:
8445:
8440:
8436:
8431:
8427:
8424:
8419:
8415:
8410:
8403:
8398:
8393:
8389:
8385:
8380:
8377:
8374:
8371:
8366:
8362:
8359:
8353:
8350:
8345:
8340:
8333:
8328:
8323:
8319:
8315:
8312:
8309:
8306:
8301:
8297:
8278:
8277:
8262:
8259:
8256:
8251:
8247:
8243:
8240:
8237:
8234:
8229:
8225:
8221:
8218:
8215:
8212:
8209:
8206:
8203:
8200:
8197:
8194:
8191:
8188:
8185:
8183:
8181:
8178:
8173:
8169:
8165:
8164:
8161:
8158:
8155:
8152:
8149:
8144:
8140:
8136:
8131:
8127:
8123:
8118:
8114:
8110:
8105:
8101:
8097:
8092:
8088:
8084:
8081:
8078:
8075:
8072:
8069:
8066:
8063:
8061:
8059:
8056:
8051:
8047:
8043:
8042:
8039:
8036:
8033:
8030:
8027:
8022:
8018:
8014:
8011:
8008:
8005:
8002:
7999:
7994:
7990:
7986:
7983:
7980:
7977:
7974:
7971:
7968:
7965:
7962:
7959:
7956:
7953:
7950:
7948:
7946:
7943:
7938:
7934:
7930:
7929:
7926:
7923:
7920:
7917:
7914:
7909:
7905:
7901:
7896:
7892:
7888:
7883:
7879:
7875:
7872:
7869:
7866:
7863:
7860:
7857:
7854:
7852:
7850:
7847:
7842:
7838:
7834:
7833:
7830:
7827:
7824:
7819:
7815:
7811:
7808:
7805:
7802:
7799:
7796:
7793:
7790:
7787:
7784:
7781:
7778:
7775:
7773:
7771:
7768:
7763:
7759:
7755:
7754:
7751:
7748:
7745:
7742:
7739:
7734:
7730:
7726:
7723:
7720:
7717:
7714:
7711:
7708:
7705:
7703:
7701:
7698:
7693:
7689:
7685:
7684:
7681:
7678:
7675:
7672:
7669:
7666:
7663:
7660:
7657:
7654:
7651:
7648:
7646:
7644:
7641:
7636:
7632:
7628:
7627:
7624:
7621:
7618:
7615:
7612:
7610:
7608:
7605:
7600:
7596:
7592:
7591:
7566:
7565:
7554:
7551:
7548:
7545:
7540:
7536:
7530:
7525:
7520:
7516:
7512:
7509:
7506:
7501:
7497:
7456:
7455:
7444:
7438:
7435:
7432:
7429:
7426:
7423:
7420:
7415:
7412:
7407:
7403:
7399:
7396:
7393:
7390:
7352:
7325:
7318:
7311:
7304:
7297:
7296:
7285:
7280:
7276:
7272:
7269:
7266:
7261:
7258:
7255:
7252:
7247:
7244:
7241:
7237:
7233:
7230:
7227:
7224:
7219:
7215:
7175:
7174:
7163:
7160:
7155:
7151:
7147:
7144:
7141:
7138:
7135:
7113:Main article:
7110:
7107:
7057:
7056:
7045:
7039:
7036:
7032:
7029:
7025:
7021:
7015:
7012:
7008:
7005:
7001:
6997:
6992:
6987:
6984:
6981:
6977:
6969:
6964:
6961:
6958:
6954:
6910:
6909:
6896:
6893:
6890:
6886:
6882:
6875:
6872:
6867:
6863:
6859:
6854:
6851:
6848:
6844:
6782:
6781:
6767:
6764:
6760:
6757:
6753:
6749:
6746:
6743:
6738:
6735:
6731:
6728:
6723:
6719:
6714:
6709:
6706:
6703:
6699:
6691:
6686:
6683:
6680:
6676:
6637:
6634:
6633:
6632:
6621:
6616:
6611:
6608:
6603:
6600:
6597:
6593:
6587:
6580:
6577:
6574:
6571:
6568:
6565:
6562:
6559:
6554:
6551:
6548:
6542:
6538:
6535:
6532:
6529:
6524:
6520:
6488:
6473:
6449:. Therefore,
6379:
6378:
6367:
6363:
6358:
6355:
6350:
6346:
6343:
6340:
6337:
6334:
6331:
6326:
6321:
6316:
6312:
6308:
6305:
6302:
6299:
6296:
6293:
6275:
6274:
6263:
6260:
6257:
6254:
6251:
6248:
6243:
6238:
6233:
6229:
6225:
6222:
6219:
6216:
6213:
6210:
6196:
6195:
6184:
6181:
6178:
6175:
6172:
6169:
6164:
6159:
6154:
6150:
6146:
6143:
6140:
6137:
6134:
6131:
6059:
6058:
6047:
6044:
6041:
6038:
6035:
6032:
6029:
6026:
6023:
6020:
6017:
5873:
5872:
5859:
5854:
5851:
5848:
5845:
5843:
5840:
5837:
5836:
5833:
5830:
5827:
5824:
5822:
5819:
5816:
5815:
5813:
5808:
5805:
5802:
5799:
5796:
5793:
5766:
5763:
5755:
5754:
5743:
5738:
5734:
5731:
5728:
5725:
5719:
5716:
5711:
5707:
5701:
5697:
5693:
5688:
5684:
5681:
5678:
5675:
5669:
5666:
5661:
5657:
5651:
5647:
5643:
5638:
5634:
5607:
5600:
5599:
5586:
5582:
5579:
5573:
5570:
5567:
5564:
5559:
5555:
5552:
5546:
5543:
5540:
5534:
5530:
5527:
5524:
5518:
5514:
5511:
5482:
5448:
5441:
5434:
5433:
5420:
5417:
5414:
5410:
5406:
5401:
5397:
5393:
5390:
5387:
5382:
5379:
5376:
5372:
5366:
5362:
5358:
5353:
5350:
5347:
5343:
5339:
5334:
5330:
5324:
5320:
5316:
5311:
5307:
5284:
5283:
5278:
5271:
5264:
5231:
5225:
5193:
5192:
5174:
5136:
5135:
5105:
5102:
5101:
5100:
5093:
5082:
5077:
5073:
5067:
5064:
5059:
5055:
5049:
5020:
5015:
4995:
4988:
4983:
4979:
4973:
4970:
4965:
4962:
4957:
4954:
4932:
4929:
4926:
4923:
4920:
4917:
4912:
4908:
4904:
4899:
4895:
4875:
4864:
4859:
4853:
4848:
4845:
4842:
4833:
4828:
4822:
4817:
4814:
4811:
4782:
4777:
4765:
4745:
4740:
4734:
4729:
4726:
4723:
4720:
4714:
4711:
4706:
4702:
4699:
4694:
4689:
4648:
4643:
4616:
4611:
4600:
4595:
4589:
4584:
4581:
4578:
4575:
4566:
4561:
4555:
4550:
4547:
4544:
4541:
4504:
4499:
4468:
4463:
4439:
4425:
4422:
4387:
4383:
4360:
4353:
4348:
4343:
4340:
4335:
4330:
4327:
4322:
4319:
4299:
4294:
4291:
4269:
4266:
4263:
4260:
4257:
4254:
4251:
4246:
4242:
4216:
4212:
4192:
4172:
4168:
4148:
4128:
4125:
4120:
4117:
4114:
4111:
4069:if and only if
4005:
4001:
3972:
3953:
3944:
3887:
3884:
3878:, is called a
3867:
3864:
3860:
3856:
3853:
3850:
3847:
3844:
3841:
3788:
3787:
3776:
3773:
3770:
3767:
3764:
3761:
3757:
3750:
3747:
3742:
3739:
3736:
3732:
3709:
3708:
3697:
3694:
3691:
3688:
3685:
3682:
3679:
3676:
3673:
3670:
3665:
3662:
3658:
3604:-sided polygon
3581:
3580:
3567:
3563:
3560:
3554:
3551:
3548:
3545:
3540:
3536:
3533:
3527:
3524:
3499:
3498:
3487:
3484:
3479:
3475:
3472:
3469:
3463:
3460:
3457:
3454:
3449:
3445:
3442:
3439:
3433:
3430:
3427:
3422:
3416:
3410:
3406:
3403:
3397:
3394:
3391:
3388:
3383:
3379:
3376:
3370:
3367:
3363:
3347:
3346:
3335:
3332:
3329:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3305:
3302:
3299:
3296:
3291:
3285:
3279:
3275:
3272:
3266:
3263:
3260:
3257:
3252:
3248:
3245:
3239:
3236:
3232:
3188:
3187:
3176:
3173:
3170:
3167:
3164:
3161:
3158:
3155:
3152:
3149:
3146:
3143:
3138:
3133:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3104:
3065:
3062:
3049:
3046:
3043:
3039:
3035:
3032:
3029:
3026:
3023:
3020:
3001:Main article:
2998:
2995:
2972:
2969:
2966:
2963:
2959:
2930:
2929:
2917:
2911:
2907:
2903:
2900:
2896:
2892:
2889:
2859:
2856:
2853:
2849:
2824:
2821:
2818:
2814:
2781:
2778:
2775:
2771:
2755:
2754:
2741:
2737:
2733:
2730:
2671:
2667:
2642:
2639:
2636:
2632:
2605:
2602:
2599:
2595:
2546:
2517:
2514:
2511:
2507:
2494:
2485:
2409:
2400:
2320:
2317:
2315:
2312:
2311:
2310:
2299:
2296:
2293:
2290:
2287:
2284:
2281:
2276:
2271:
2266:
2262:
2230:, and that of
2181:
2180:
2169:
2166:
2163:
2160:
2157:
2154:
2149:
2144:
2139:
2135:
2131:
2128:
2125:
2122:
2119:
2116:
2094:disjoint union
1992:
1991:
1980:
1974:
1971:
1968:
1965:
1962:
1959:
1955:
1950:
1944:
1941:
1938:
1935:
1932:
1929:
1926:
1921:
1918:
1912:
1907:
1903:
1900:
1897:
1894:
1891:
1888:
1885:
1879:
1876:
1809:
1806:
1803:
1800:
1797:
1794:
1783:
1782:
1771:
1765:
1762:
1759:
1756:
1753:
1750:
1746:
1741:
1738:
1681:
1677:
1674:
1670:
1667:
1662:
1659:
1656:
1636:
1633:
1628:
1624:
1620:
1615:
1611:
1601:, and one has
1570:
1565:
1561:
1557:
1552:
1548:
1526:
1523:
1519:
1516:
1511:
1508:
1505:
1479:
1475:
1472:
1468:
1465:
1460:
1457:
1454:
1444:if and only if
1430:
1426:
1422:
1417:
1413:
1199:
1198:
1187:
1182:
1178:
1174:
1169:
1165:
1159:
1155:
1151:
1146:
1142:
1136:
1132:
1128:
1123:
1119:
1115:
1110:
1107:
1103:
1097:
1093:
1089:
1084:
1081:
1078:
1075:
1071:
1067:
1062:
1058:
979:
978:
967:
961:
958:
952:
947:
944:
941:
937:
933:
928:
925:
921:
917:
912:
909:
873:
872:
861:
858:
853:
849:
845:
840:
836:
830:
826:
822:
819:
814:
811:
807:
803:
798:
794:
788:
784:
780:
715:
712:
657:
656:
645:
642:
639:
636:
633:
630:
627:
624:
621:
618:
615:
612:
609:
601:
598:
593:
589:
578:
575:
570:
566:
507:characteristic
460:
457:
454:
451:
448:
445:
442:
439:
436:
433:
430:
426:
421:
417:
414:
411:
405:
402:
399:
396:
391:
387:
384:
381:
375:
372:
369:
365:
360:
356:
353:
350:
347:
341:
337:
334:
319:imaginary part
294:
291:
286:
282:
231:
228:
197:characteristic
162:complex number
128:
127:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
10952:
10941:
10938:
10936:
10933:
10931:
10928:
10926:
10923:
10921:
10918:
10917:
10915:
10902:
10901:
10896:
10888:
10882:
10879:
10877:
10870:
10868:
10861:
10859:
10856:
10854:
10851:
10849:
10846:
10844:
10841:
10839:
10836:
10834:
10822:
10820:
10817:
10815:
10814:Root of unity
10812:
10810:
10807:
10805:
10802:
10800:
10793:
10791:
10788:
10786:
10785:Perron number
10783:
10781:
10774:
10772:
10769:
10767:
10764:
10762:
10759:
10757:
10754:
10752:
10749:
10747:
10744:
10742:
10739:
10737:
10734:
10733:
10730:
10726:
10719:
10714:
10712:
10707:
10705:
10700:
10699:
10696:
10688:
10686:0-309-09657-X
10682:
10678:
10673:
10672:
10666:
10662:
10658:
10656:0-387-94762-0
10652:
10648:
10644:
10640:
10636:
10634:3-540-15251-2
10630:
10626:
10621:
10617:
10613:
10609:
10605:
10601:
10595:
10591:
10586:
10581:
10577:
10573:
10569:
10565:
10560:
10556:
10552:
10548:
10544:
10541:
10537:
10533:
10529:
10525:
10519:
10515:
10511:
10507:
10503:
10502:
10498:
10490:
10486:
10480:
10477:
10472:
10468:
10464:
10460:
10456:
10452:
10448:
10444:
10438:
10435:
10430:
10423:
10420:
10405:
10398:
10395:
10390:
10388:0-300-09473-6
10384:
10380:
10376:
10370:
10367:
10362:
10358:
10354:
10350:
10346:
10342:
10341:Landau, Susan
10336:
10333:
10327:
10322:
10318:
10314:
10310:
10306:
10300:
10297:
10292:
10286:
10282:
10278:
10274:
10273:
10268:
10262:
10259:
10254:
10252:0-8176-3743-5
10248:
10244:
10243:
10238:
10232:
10230:
10228:
10224:
10219:
10217:9781470415549
10213:
10209:
10208:
10203:
10197:
10194:
10189:
10183:
10179:
10178:
10170:
10167:
10162:
10156:
10152:
10148:
10144:
10143:
10135:
10133:
10129:
10124:
10118:
10114:
10110:
10106:
10105:
10100:
10093:
10090:
10085:
10083:9789812794949
10079:
10075:
10074:
10066:
10063:
10058:
10051:
10048:
10043:
10037:
10033:
10029:
10022:
10019:
10014:
10008:
10004:
10003:
9998:
9992:
9989:
9982:
9978:
9975:
9973:
9970:
9968:
9965:
9963:
9960:
9958:
9955:
9953:
9950:
9947:
9944:
9942:
9941:Argand system
9939:
9938:
9934:
9932:
9928:
9923:
9913:
9907:
9900:
9894:
9890:
9885:
9875:
9869:
9862:
9856:
9854:
9850:
9847:
9841:
9836:
9834:
9829:
9824:
9821:
9817:
9811:
9806:
9800:
9794:
9788:
9776:
9770:
9765:
9759:
9754:
9749:
9748:
9742:
9736:
9731:
9730:
9729:
9727:
9718:
9716:
9712:
9708:
9702:
9692:
9685:
9684:complex plane
9680:
9673:
9671:
9669:
9668:
9663:
9659:
9654:
9652:
9648:
9647:
9642:
9638:
9637:Galois theory
9634:
9630:
9626:
9622:
9618:
9614:
9592:
9582:
9578:
9574:
9571:
9568:
9562:
9559:
9542:
9529:
9521:
9517:
9503:
9499:
9496:
9492:
9488:
9485:has degree φ(
9466:
9456:
9452:
9448:
9445:
9442:
9436:
9433:
9418:
9402:
9385:
9378:contains all
9377:
9361:
9352:
9348:
9344:
9341:
9338:
9332:
9329:
9314:
9294:
9275:
9267:
9265:
9263:
9259:
9255:
9251:
9247:
9243:
9238:
9236:
9232:
9224:
9215:
9209:
9205:
9201:
9194:
9190:
9179:Cyclic groups
9178:
9176:
9174:
9170:
9158:
9153:
9149:
9145:
9138:
9133:
9128:
9118:
9111:
9106:is odd, then
9105:
9089:
9084:
9080:
9076:
9071:
9067:
9063:
9058:
9054:
9031:
9027:
9023:
9020:
9017:
9012:
9008:
9004:
8999:
8995:
8974:
8969:
8965:
8961:
8956:
8952:
8946:
8942:
8938:
8935:
8927:
8909:
8896:
8886:
8879:
8874:
8872:
8869:
8846:
8841:
8837:
8831:
8828:
8825:
8820:
8817:
8814:
8810:
8806:
8800:
8797:
8794:
8789:
8786:
8781:
8777:
8770:
8764:
8756:
8735:
8722:
8718:
8714:
8710:
8701:
8697:
8693:
8689:
8685:
8681:
8677:
8673:
8669:
8665:
8656:
8652:
8648:
8644:
8640:
8636:
8632:
8628:
8619:
8615:
8611:
8607:
8603:
8599:
8595:
8591:
8582:
8578:
8574:
8570:
8566:
8557:
8553:
8549:
8545:
8541:
8537:
8528:
8524:
8520:
8516:
8512:
8503:
8499:
8495:
8486:
8485:
8484:
8482:
8477:
8457:
8451:
8446:
8443:
8438:
8434:
8429:
8425:
8422:
8417:
8413:
8408:
8401:
8391:
8387:
8383:
8375:
8369:
8364:
8360:
8357:
8351:
8348:
8343:
8338:
8331:
8321:
8317:
8313:
8307:
8299:
8287:
8286:
8285:
8283:
8257:
8254:
8249:
8245:
8235:
8232:
8227:
8223:
8213:
8210:
8207:
8198:
8195:
8192:
8186:
8184:
8179:
8176:
8171:
8167:
8156:
8153:
8150:
8147:
8142:
8138:
8134:
8129:
8125:
8121:
8116:
8112:
8108:
8103:
8099:
8095:
8090:
8086:
8076:
8073:
8070:
8064:
8062:
8057:
8054:
8049:
8045:
8034:
8031:
8028:
8025:
8020:
8016:
8006:
8003:
8000:
7997:
7992:
7988:
7978:
7975:
7972:
7963:
7960:
7957:
7951:
7949:
7944:
7941:
7936:
7932:
7921:
7918:
7915:
7912:
7907:
7903:
7899:
7894:
7890:
7886:
7881:
7877:
7867:
7864:
7861:
7855:
7853:
7848:
7845:
7840:
7836:
7825:
7822:
7817:
7813:
7803:
7800:
7797:
7788:
7785:
7782:
7776:
7774:
7769:
7766:
7761:
7757:
7746:
7743:
7740:
7737:
7732:
7728:
7718:
7715:
7712:
7706:
7704:
7699:
7696:
7691:
7687:
7676:
7673:
7670:
7661:
7658:
7655:
7649:
7647:
7642:
7639:
7634:
7630:
7622:
7619:
7616:
7613:
7611:
7606:
7603:
7598:
7594:
7582:
7581:
7580:
7576:
7571:
7570:factorization
7552:
7546:
7538:
7528:
7518:
7514:
7510:
7507:
7504:
7499:
7495:
7487:
7486:
7485:
7476:
7463:
7461:
7442:
7436:
7433:
7427:
7424:
7421:
7413:
7410:
7405:
7397:
7394:
7391:
7378:
7377:
7376:
7374:
7366:
7360:
7355:
7348:
7342:
7329:
7324:
7317:
7310:
7303:
7278:
7274:
7270:
7267:
7256:
7250:
7245:
7242:
7239:
7235:
7231:
7225:
7217:
7205:
7204:
7203:
7198:
7194:
7193:
7184:
7161:
7158:
7153:
7149:
7145:
7139:
7133:
7126:
7125:
7124:
7122:
7116:
7108:
7106:
7102:
7098:
7094:
7089:
7083:
7079:
7070:
7066:
7043:
7037:
7034:
7030:
7027:
7023:
7019:
7013:
7010:
7006:
7003:
6999:
6995:
6985:
6982:
6979:
6975:
6967:
6962:
6959:
6956:
6952:
6944:
6943:
6942:
6940:
6930:
6926:
6925:
6919:
6915:
6894:
6891:
6888:
6884:
6880:
6873:
6870:
6865:
6861:
6857:
6852:
6849:
6846:
6842:
6834:
6833:
6832:
6828:
6824:
6815:
6811:
6807:
6801:
6791:
6765:
6762:
6758:
6755:
6751:
6747:
6744:
6741:
6736:
6733:
6729:
6726:
6721:
6717:
6707:
6704:
6701:
6697:
6689:
6684:
6681:
6678:
6674:
6666:
6665:
6664:
6662:
6658:
6652:
6648:
6643:
6642:orthogonality
6636:Orthogonality
6635:
6619:
6614:
6609:
6606:
6601:
6598:
6595:
6591:
6585:
6578:
6575:
6569:
6566:
6563:
6552:
6549:
6546:
6540:
6536:
6530:
6522:
6518:
6510:
6509:
6508:
6496:
6491:
6487:
6483:
6476:
6472:
6466:
6462:
6458:
6454:
6442:
6434:
6423:
6419:
6408:
6400:
6389:
6385:
6365:
6361:
6356:
6353:
6348:
6344:
6341:
6335:
6329:
6324:
6314:
6310:
6306:
6300:
6294:
6291:
6284:
6283:
6282:
6280:
6277:Applying the
6261:
6255:
6249:
6246:
6241:
6231:
6227:
6223:
6217:
6211:
6208:
6201:
6200:
6199:
6198:This implies
6182:
6176:
6170:
6162:
6152:
6148:
6144:
6138:
6132:
6122:
6121:
6120:
6116:
6108:
6100:
6088:
6082:
6077:
6075:
6069:
6065:
6045:
6039:
6033:
6030:
6024:
6018:
6015:
6008:
6007:
6006:
5998:
5991:
5987:
5979:
5975:
5971:
5965:
5961:
5952:
5943:
5936:
5929:
5922:
5916:
5912:
5905:
5898:
5893:
5887:
5878:
5852:
5849:
5846:
5841:
5838:
5831:
5828:
5825:
5820:
5817:
5811:
5806:
5800:
5794:
5791:
5784:
5783:
5782:
5774:
5764:
5762:
5760:
5741:
5736:
5732:
5729:
5726:
5723:
5717:
5714:
5709:
5705:
5699:
5695:
5691:
5686:
5682:
5679:
5676:
5673:
5667:
5664:
5659:
5655:
5649:
5645:
5641:
5636:
5632:
5624:
5623:
5622:
5610:
5606:
5584:
5580:
5577:
5571:
5568:
5565:
5562:
5557:
5553:
5550:
5544:
5541:
5538:
5532:
5528:
5525:
5522:
5516:
5512:
5509:
5502:
5501:
5500:
5493:
5491:
5488:is a complex
5485:
5481:
5476:
5464:
5459:
5451:
5447:
5440:
5418:
5415:
5412:
5408:
5404:
5399:
5395:
5391:
5388:
5385:
5380:
5377:
5374:
5370:
5364:
5360:
5356:
5351:
5348:
5345:
5341:
5337:
5332:
5328:
5322:
5318:
5314:
5309:
5305:
5297:
5296:
5295:
5289:
5277:
5270:
5263:
5258:
5257:
5256:
5251:
5243:
5239:
5234:
5224:
5217:
5213:
5203:
5199:
5189:
5185:
5181:
5177:
5170:
5169:
5168:
5157:
5153:
5150:
5146:
5132:
5128:
5124:
5119:
5118:
5117:
5103:
5098:
5094:
5080:
5075:
5071:
5065:
5062:
5057:
5053:
5047:
5038:
5030:
5026:
5016:
5013:
4993:
4986:
4981:
4977:
4971:
4968:
4963:
4960:
4955:
4952:
4930:
4927:
4924:
4921:
4918:
4915:
4910:
4906:
4902:
4897:
4893:
4884:
4880:
4876:
4862:
4857:
4851:
4846:
4843:
4840:
4831:
4826:
4820:
4815:
4812:
4809:
4796:
4792:
4788:
4778:
4763:
4743:
4738:
4732:
4727:
4724:
4721:
4718:
4712:
4709:
4704:
4700:
4697:
4692:
4687:
4676:
4670:
4666:
4662:
4658:
4654:
4644:
4640:
4633:
4626:
4622:
4612:
4598:
4593:
4587:
4582:
4579:
4576:
4573:
4564:
4559:
4553:
4548:
4545:
4542:
4539:
4528:
4524:
4518:
4514:
4510:
4500:
4497:
4491:
4485:
4478:
4474:
4464:
4461:
4456:
4449:
4445:
4433:
4428:
4427:
4423:
4421:
4419:
4415:
4407:
4385:
4381:
4371:
4358:
4351:
4346:
4341:
4338:
4333:
4328:
4325:
4320:
4317:
4297:
4292:
4289:
4267:
4264:
4261:
4258:
4255:
4252:
4249:
4244:
4240:
4232:
4214:
4210:
4195:
4170:
4166:
4157:
4151:
4126:
4123:
4118:
4115:
4112:
4109:
4100:
4085:
4083:
4082:Fermat primes
4079:
4074:
4070:
4066:
4058:
4054:
4046:
4041:
4039:
4003:
3999:
3989:
3988:Galois theory
3985:
3982:
3975:
3967:
3962:
3956:
3947:
3940:
3935:
3929:
3922:
3917:
3913:
3907:
3903:
3899:
3894:
3885:
3883:
3881:
3862:
3858:
3854:
3851:
3848:
3842:
3839:
3831:
3827:
3810:
3802:
3774:
3771:
3768:
3765:
3762:
3759:
3755:
3748:
3745:
3740:
3737:
3734:
3730:
3722:
3721:
3720:
3695:
3692:
3689:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3663:
3660:
3656:
3648:
3647:
3646:
3645:
3641:
3639:
3635:
3631:
3627:
3621:
3614:
3609:
3605:
3593:
3592:complex plane
3588:
3565:
3561:
3558:
3552:
3549:
3546:
3543:
3538:
3534:
3531:
3525:
3522:
3515:
3514:
3513:
3509:
3505:
3485:
3482:
3477:
3473:
3470:
3467:
3461:
3458:
3455:
3452:
3447:
3443:
3440:
3437:
3431:
3428:
3425:
3420:
3414:
3408:
3404:
3401:
3395:
3392:
3389:
3386:
3381:
3377:
3374:
3368:
3365:
3361:
3352:
3351:
3350:
3333:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3303:
3300:
3297:
3294:
3289:
3283:
3277:
3273:
3270:
3264:
3261:
3258:
3255:
3250:
3246:
3243:
3237:
3234:
3230:
3221:
3220:
3219:
3208:
3194:
3174:
3171:
3168:
3165:
3162:
3159:
3156:
3153:
3150:
3147:
3144:
3141:
3136:
3131:
3127:
3124:
3121:
3118:
3115:
3112:
3109:
3106:
3102:
3093:
3092:
3091:
3086:and integers
3082:
3078:
3070:
3063:
3061:
3047:
3041:
3037:
3033:
3030:
3024:
3021:
3018:
3010:
3004:
2996:
2994:
2992:
2988:
2983:
2970:
2964:
2948:
2946:
2939:
2935:
2915:
2909:
2905:
2898:
2894:
2887:
2880:
2879:
2878:
2876:
2871:
2854:
2838:
2819:
2802:
2796:
2776:
2760:
2739:
2735:
2728:
2721:
2720:
2719:
2717:
2712:
2706:
2700:
2694:
2688:
2682:
2669:
2656:
2637:
2620:
2616:contains all
2600:
2585:
2580:
2574:
2568:
2562:
2535:
2531:
2512:
2491:
2486:
2484:
2482:
2478:
2474:
2470:
2465:
2459:
2453:
2447:
2441:
2435:
2432:
2426:
2420:
2415:
2406:
2401:
2399:
2397:
2393:
2389:
2385:
2384:abelian group
2380:
2377:
2371:
2366:
2361:
2354:
2346:
2338:
2331:
2326:
2318:
2313:
2297:
2294:
2291:
2285:
2279:
2274:
2264:
2260:
2252:
2251:
2250:
2246:
2242:
2235:
2228:
2221:
2215:
2210:
2207:
2197:
2192:
2187:
2167:
2161:
2155:
2147:
2137:
2133:
2129:
2123:
2117:
2107:
2106:
2105:
2101:
2095:
2089:
2081:
2074:
2067:
2060:
2053:
2048:
2043:
2037:
2030:
2026:
2020:
2014:
2009:
2004:
1998:
1978:
1969:
1966:
1963:
1953:
1948:
1939:
1936:
1933:
1924:
1919:
1916:
1910:
1905:
1898:
1895:
1892:
1886:
1883:
1877:
1874:
1867:
1866:
1865:
1855:
1850:
1836:
1823:
1804:
1801:
1798:
1769:
1760:
1757:
1754:
1744:
1739:
1736:
1729:
1728:
1727:
1724:
1714:
1710:
1704:
1698:
1692:
1679:
1672:
1668:
1660:
1657:
1654:
1634:
1631:
1626:
1622:
1618:
1613:
1609:
1598:
1592:
1585:
1568:
1563:
1559:
1555:
1550:
1546:
1521:
1517:
1509:
1506:
1503:
1494:
1477:
1470:
1466:
1458:
1455:
1452:
1445:
1428:
1424:
1420:
1415:
1411:
1401:
1395:
1389:
1386:
1381:
1378:
1373:
1367:
1360:
1356:
1350:
1344:
1338:
1332:
1325:
1319:
1315:
1311:
1304:
1300:
1294:
1287:
1283:
1277:
1271:
1265:
1259:
1253:
1247:
1237:
1232:
1228:
1221:
1217:
1211:
1205:
1185:
1180:
1176:
1172:
1167:
1163:
1157:
1153:
1149:
1144:
1134:
1130:
1121:
1117:
1113:
1108:
1105:
1101:
1095:
1091:
1087:
1082:
1079:
1076:
1073:
1069:
1065:
1060:
1056:
1048:
1047:
1046:
1043:
1037:
1033:
1029:
1024:
1023:
1016:
1012:
1005:
1001:
997:
991:
985:
965:
956:
950:
945:
942:
939:
935:
931:
926:
923:
919:
915:
910:
907:
898:
897:
896:
893:
888:
883:
878:
859:
856:
851:
847:
843:
838:
828:
824:
817:
812:
809:
805:
801:
796:
786:
782:
771:
770:
769:
766:
760:
754:
750:
744:
740:
734:
728:
722:
713:
711:
709:
708:
702:
698:
693:
691:
674:
671:
666:
662:
643:
640:
637:
634:
631:
628:
625:
622:
619:
616:
613:
610:
607:
599:
596:
591:
587:
576:
573:
568:
564:
556:
555:
554:
553:, that is if
544:
531:
529:
525:
524:
518:
513:
508:
503:
497:
491:
485:
480:
476:
471:
458:
455:
452:
449:
446:
443:
440:
437:
434:
431:
428:
424:
419:
415:
412:
409:
403:
400:
397:
394:
389:
385:
382:
379:
373:
370:
367:
363:
358:
354:
351:
348:
345:
339:
335:
332:
320:
316:
308:
292:
289:
284:
280:
272:
260:
241:
236:
229:
227:
214:
210:
206:
202:
198:
194:
189:
187:
183:
179:
178:number theory
171:
167:
163:
159:
157:
152:
151:root of unity
148:
141:
140:complex plane
136:
132:
124:
121:
113:
102:
99:
95:
92:
88:
85:
81:
78:
74:
71: –
70:
66:
65:Find sources:
59:
55:
49:
48:
43:This article
41:
37:
32:
31:
19:
10891:
10819:Salem number
10813:
10670:
10646:
10624:
10579:
10568:Course Notes
10567:
10555:Course Notes
10554:
10509:
10484:
10479:
10454:
10450:
10437:
10428:
10422:
10410:. Retrieved
10397:
10378:
10369:
10352:
10348:
10335:
10316:
10312:
10305:Lehmer, Emma
10299:
10271:
10261:
10241:
10237:Riesel, Hans
10206:
10196:
10176:
10169:
10141:
10103:
10099:Pilz, Günter
10092:
10072:
10065:
10056:
10050:
10031:
10021:
10001:
9991:
9946:Circle group
9926:
9909:
9905:
9898:
9895:
9888:
9871:
9867:
9860:
9857:
9849:golden ratio
9839:
9832:
9822:
9813:
9809:
9798:
9795:
9780:
9768:
9757:
9746:
9734:
9719:
9700:
9697:
9665:
9657:
9656:Conversely,
9655:
9644:
9632:
9628:
9624:
9620:
9543:
9504:of the ring
9491:Galois group
9486:
9277:
9242:eigenvectors
9239:
9216:
9189:cyclic group
9182:
9154:
9147:
9143:
9136:
9131:
9126:
9119:
9109:
9103:
8907:
8877:
8875:
8734:prime number
8727:
8720:
8716:
8712:
8708:
8695:
8691:
8687:
8683:
8679:
8675:
8671:
8667:
8663:
8650:
8646:
8642:
8638:
8634:
8630:
8626:
8613:
8609:
8605:
8601:
8597:
8593:
8589:
8576:
8572:
8568:
8564:
8551:
8547:
8543:
8539:
8535:
8522:
8518:
8514:
8510:
8497:
8493:
8475:
8472:
8279:
7574:
7567:
7464:
7457:
7358:
7353:
7340:
7327:
7322:
7315:
7308:
7301:
7298:
7196:
7190:
7183:multiplicity
7176:
7118:
7100:
7096:
7092:
7081:
7077:
7058:
6941:. That is,
6928:
6922:
6911:
6831:th entry is
6826:
6822:
6809:
6805:
6802:
6783:
6660:
6656:
6650:
6646:
6639:
6494:
6489:
6485:
6474:
6470:
6467:
6460:
6456:
6452:
6440:
6432:
6421:
6417:
6406:
6398:
6387:
6383:
6380:
6276:
6197:
6114:
6106:
6098:
6086:
6078:
6067:
6063:
6060:
5996:
5992:
5985:
5977:
5973:
5969:
5963:
5955:
5941:
5934:
5927:
5920:
5917:
5910:
5903:
5896:
5885:
5874:
5772:
5768:
5756:
5608:
5604:
5601:
5494:
5483:
5479:
5460:
5449:
5445:
5438:
5435:
5285:
5275:
5268:
5261:
5249:
5242:linear space
5229:
5222:
5215:
5211:
5201:
5197:
5194:
5187:
5183:
5179:
5172:
5155:
5151:
5148:
5144:
5137:
5130:
5126:
5122:
5107:
5097:Heptadecagon
5036:
5028:
5024:
4794:
4790:
4786:
4668:
4664:
4660:
4656:
4652:
4638:
4631:
4624:
4620:
4516:
4512:
4508:
4489:
4483:
4476:
4472:
4459:
4454:
4447:
4443:
4431:
4417:
4372:
4193:
4149:
4098:
4086:
4078:power of two
4076:is either a
4072:
4053:square roots
4042:
3986:
3980:
3973:
3965:
3959:is given by
3954:
3945:
3933:
3927:
3920:
3918:(except for
3905:
3892:
3889:
3808:
3800:
3789:
3710:
3642:
3619:
3612:
3589:
3582:
3507:
3503:
3500:
3348:
3206:
3192:
3189:
3075:
3006:
2984:
2944:
2936:between the
2931:
2872:
2837:Galois group
2800:
2794:
2759:automorphism
2756:
2710:
2704:
2698:
2692:
2686:
2683:
2618:
2578:
2572:
2566:
2560:
2496:
2489:
2481:circle group
2473:cyclic group
2472:
2469:cyclic group
2463:
2457:
2451:
2449:, the other
2445:
2439:
2436:
2430:
2424:
2418:
2413:
2411:
2404:
2396:circle group
2381:
2375:
2369:
2359:
2352:
2344:
2336:
2329:
2322:
2244:
2240:
2233:
2226:
2219:
2211:
2205:
2199:, including
2195:
2185:
2182:
2099:
2087:
2079:
2072:
2065:
2061:
2051:
2041:
2035:
2028:
2024:
2018:
2012:
2002:
1996:
1993:
1848:
1834:
1784:
1722:
1712:
1708:
1702:
1696:
1693:
1596:
1590:
1583:
1492:
1399:
1393:
1390:
1384:
1371:
1365:
1358:
1354:
1348:
1342:
1336:
1330:
1323:
1317:
1313:
1309:
1302:
1298:
1292:
1285:
1281:
1275:
1269:
1263:
1257:
1251:
1248:
1230:
1226:
1219:
1215:
1209:
1203:
1200:
1045:, and hence
1041:
1035:
1031:
1027:
1021:
1014:
1010:
1003:
999:
995:
989:
983:
980:
891:
881:
874:
764:
758:
755:
748:
742:
738:
732:
726:
720:
717:
706:
694:
675:
669:
665:prime number
660:
658:
538:
532:
528:Finite field
522:
517:finite field
511:
501:
495:
489:
483:
472:
255:
253:
239:
205:finite field
190:
154:
150:
144:
131:
116:
107:
97:
90:
83:
76:
64:
52:Please help
47:verification
44:
10930:Polynomials
10506:Lang, Serge
10451:SIAM Review
10431:. Springer.
9972:Witt vector
8912:) then the
5945:of all the
5892:coefficient
5163:), and the
5104:Periodicity
3916:irreducible
3608:unit circle
3506:= 1, 2, …,
2875:composition
2757:induces an
2570:. As every
2214:cardinality
1647:, although
1388:solutions.
667:, then all
147:mathematics
10935:1 (number)
10914:Categories
10616:0956.11021
10540:0984.00001
10499:References
9498:isomorphic
9489:) and its
9173:algorithms
6912:defines a
6415:, and for
5894:of degree
5757:This is a
4879:cube roots
4040:, below.)
3902:polynomial
2932:defines a
2212:Since the
1213:, one has
877:reciprocal
209:conversely
184:, and the
110:April 2012
80:newspapers
9789:(also an
9762:they are
9739:they are
9662:Kronecker
9572:π
9563:
9495:naturally
9446:π
9437:
9342:π
9333:
9262:Laplacian
9208:generator
9146:≡ 1 (mod
9021:⋯
8962:⋯
8829:−
8811:∑
8798:−
8787:−
8753:Φ
8435:μ
8423:−
8388:∏
8370:μ
8358:−
8318:∏
8296:Φ
8280:Applying
8196:−
8177:−
8074:−
8055:−
8026:−
7961:−
7942:−
7865:−
7846:−
7786:−
7767:−
7716:−
7697:−
7659:−
7640:−
7620:−
7604:−
7535:Φ
7515:∏
7505:−
7434:−
7411:−
7271:−
7251:φ
7236:∏
7214:Φ
7197:primitive
7159:−
7024:δ
6996:⋅
6991:¯
6953:∑
6920:requires
6892:⋅
6881:⋅
6866:−
6752:δ
6748:⋅
6734:⋅
6718:⋅
6713:¯
6705:⋅
6675:∑
6599:π
6541:∑
6345:
6330:μ
6311:∑
6295:
6250:
6228:∑
6212:
6171:
6149:⋃
6133:
6034:μ
6019:
5982:, whence
5795:
5765:Summation
5727:π
5718:
5696:∑
5677:π
5668:
5646:∑
5581:π
5572:
5554:π
5545:
5526:π
5490:amplitude
5475:frequency
5416:⋅
5405:⋅
5389:⋯
5378:⋅
5349:⋅
5338:⋅
5319:∑
5200:= 1, … ,
5063:±
5048:±
4972:−
4961:±
4925:−
4916:−
4844:−
4764:ε
4728:ε
4710:±
4698:−
4688:ε
4580:−
4574:−
4540:−
4329:−
4318:±
4250:−
4185:that has
3852:π
3843:
3830:real part
3763:≤
3738:π
3690:
3675:
3562:π
3553:
3535:π
3526:
3483:≠
3474:π
3462:
3444:π
3432:
3405:π
3396:
3378:π
3369:
3325:π
3319:
3307:π
3301:
3274:π
3265:
3247:π
3238:
3166:
3148:
3125:
3110:
3034:π
3025:
2965:ω
2906:ω
2902:↦
2899:ω
2891:↦
2855:ω
2820:ω
2777:ω
2736:ω
2732:↦
2729:ω
2638:ω
2601:ω
2513:ω
2402:Group of
2280:φ
2261:∑
2156:
2134:⋃
2118:
1994:Thus, if
1887:
1507:≡
1456:≡
960:¯
943:−
924:−
641:−
632:…
597:≠
541:primitive
456:−
447:…
416:π
404:
386:π
374:
352:π
336:
195:. If the
160:, is any
156:de Moivre
10645:(1997).
10578:(1999).
10549:(1998).
10508:(2002),
10445:(1999).
10377:(1965).
10307:(1936).
10269:(1976).
10239:(1994).
10204:(2015).
10101:(1984).
9999:(2000).
9935:See also
9715:integers
9617:subfield
9615:. Every
9157:radicals
8674:− 1) =
8645:− 1) =
7038:′
7014:′
6766:′
6730:′
5976:) = SR(
5205:are all
5035:±
4061:regular
3600:regular
3190:Setting
2991:radicals
2477:subgroup
2357:, where
2191:divisors
1864:. Thus
1658:≢
1538:implies
1223:, where
998:≡
271:equation
261:, where
10608:1697859
10532:1878556
10510:Algebra
10459:Bibcode
10412:22 June
10032:Algebra
9918:√
9880:√
9846:inverse
9801:= 5, 10
9773:): see
9682:In the
9386:of the
9244:of any
8871:repunit
8719:− 1) =
8600:− 1) =
8575:− 1) =
8546:− 1) =
8521:− 1) =
8479:is the
7475:divisor
7185:1. The
6939:unitary
6788:is the
6445:
6429:
6411:
6395:
6391:, then
6072:is the
5958:
5602:allows
5244:of all
5240:of the
4462:> 1.
3900:of the
3813:
3797:
3211:
3197:
2987:abelian
2716:coprime
2532:of the
2528:be the
2479:of the
2394:of the
2390:is the
2363:is the
2341:, then
2096:of the
2008:coprime
1852:is the
1820:is the
1346:, ...,
1321:, then
1273:, ...,
207:, and,
170:integer
94:scholar
10683:
10653:
10631:
10614:
10606:
10596:
10538:
10530:
10520:
10489:Galois
10385:
10287:
10249:
10214:
10184:
10157:
10119:
10080:
10038:
10009:
9737:= 3, 6
9703:= 1, 2
9200:finite
8987:where
8715:− 1)⋅(
8670:− 1)⋅(
8641:− 1)⋅(
8637:− 1)⋅(
8633:− 1)⋅(
8596:− 1)⋅(
8571:− 1)⋅(
8542:− 1)⋅(
8517:− 1)⋅(
8473:where
7465:Every
7299:where
6819:whose
6814:matrix
6786:δ
6784:where
6281:gives
6061:where
5930:> 1
5913:> 1
5444:, … ,
5228:, … ,
5178:: … ,
5006:where
4835:
4756:where
4568:
4529:, are
4492:> 2
4102:, and
4045:proved
4043:Gauss
2582:, the
2349:, and
1785:where
1377:degree
1306:where
987:is an
879:of an
718:Every
703:, see
248:φ
172:power
166:raised
158:number
96:
89:
82:
75:
67:
10407:(PDF)
9983:Notes
9853:minus
9658:every
9651:Gauss
9376:field
9374:This
9193:order
9169:Gauss
8865:base
8732:is a
8711:) = (
8666:) = (
8629:) = (
8592:) = (
8567:) = (
8538:) = (
8513:) = (
7321:, …,
7121:zeros
7065:Gauss
6447:) = 1
6413:) = 0
6386:<
5988:) = 0
5951:group
5473:is a
5465:. If
5238:basis
4406:empty
4154:is a
3898:roots
3638:tomos
3634:cyclo
3090:, is
2938:units
2653:is a
2584:field
2388:group
2355:) = 1
2347:) = 1
2092:is a
1312:<
1229:<
1008:then
1002:(mod
663:is a
475:field
193:field
101:JSTOR
87:books
10681:ISBN
10651:ISBN
10629:ISBN
10594:ISBN
10518:ISBN
10483:The
10414:2007
10383:ISBN
10285:ISBN
10247:ISBN
10212:ISBN
10182:ISBN
10155:ISBN
10117:ISBN
10078:ISBN
10036:ISBN
10007:ISBN
9901:= 12
9896:For
9858:For
9851:and
9828:ring
9818:= 2
9796:For
9771:= −1
9755:For
9750:= −3
9732:For
9713:are
9709:and
9698:For
9415:The
9217:The
9183:The
9108:1 −
9102:and
9077:>
9024:<
9018:<
9005:<
8496:) =
7119:The
7099:log
6803:The
6792:and
6654:and
6455:) =
5993:Let
5850:>
5769:Let
5617:and
5477:and
5260:… ,
5195:for
5121:… ,
5095:See
5027:) =
4789:) =
4655:) =
4635:and
4623:) =
4523:cube
4511:) =
4496:real
4475:) =
4446:) =
4429:For
4065:-gon
4059:the
3890:The
3820:and
3769:<
3628:and
3617:and
3594:the
3501:for
3349:but
3081:real
2714:are
2708:and
2497:Let
2373:and
2334:and
2203:and
2006:are
2000:and
1860:and
1828:and
1694:Let
1599:= –1
1308:1 ≤
1290:are
1249:Let
1225:0 ≤
688:are
684:and
526:and
479:ring
315:even
149:, a
73:news
10612:Zbl
10536:Zbl
10467:doi
10357:doi
10321:doi
10277:doi
10147:doi
10109:doi
9931:).
9929:= 3
9893:).
9891:= 2
9863:= 8
9842:= 5
9760:= 4
9649:of
9560:exp
9493:is
9434:exp
9330:exp
9311:th
9191:of
8895:odd
8885:105
8878:not
8728:If
8723:+ 1
8698:+ 1
8653:+ 1
8616:+ 1
8579:+ 1
8554:+ 1
8525:+ 1
8500:− 1
7577:− 1
7480:of
7345:is
7189:th
6937:is
6558:gcd
6480:of
6478:(1)
6451:SP(
6427:SR(
6393:SR(
5995:SP(
5984:SR(
5972:SR(
5962:=
5937:≠ 1
5923:= 1
5908:or
5906:= 1
5899:– 1
5888:– 1
5771:SR(
5715:sin
5665:cos
5619:sin
5615:cos
5569:sin
5542:cos
5281:, …
5250:any
5190:, …
5138:is
5133:, …
5108:If
5031:+ 1
5017:As
4797:+ 1
4779:As
4671:+ 1
4645:As
4627:+ 1
4613:As
4519:+ 1
4501:As
4479:+ 1
4465:As
4450:− 1
4434:= 1
4087:If
3937:th
3923:= 1
3908:− 1
3840:cos
3687:sin
3672:cos
3622:= 5
3615:= 3
3550:sin
3523:cos
3510:− 1
3459:sin
3429:cos
3393:sin
3366:cos
3316:sin
3298:cos
3262:sin
3235:cos
3163:sin
3145:cos
3122:sin
3107:cos
3022:cos
3011:of
2839:of
2761:of
2684:If
2657:of
2367:of
2339:= 1
2332:= 1
2238:is
2224:is
2216:of
2193:of
2045:is
1958:gcd
1928:gcd
1884:lcm
1856:of
1824:of
1793:gcd
1749:gcd
1716:of
1669:mod
1586:= 4
1518:mod
1490:If
1467:mod
1375:th-
1361:= 1
1326:= 1
1288:= 1
1242:by
1238:of
1207:of
981:If
751:= 1
659:If
582:and
533:An
509:of
481:)
401:sin
371:cos
333:exp
313:is
254:An
145:In
56:by
10916::
10679:.
10610:.
10604:MR
10602:.
10592:.
10582:.
10566:.
10553:.
10534:,
10528:MR
10526:,
10512:,
10465:.
10455:41
10453:.
10449:.
10353:30
10351:.
10343:;
10317:42
10315:.
10311:.
10283:.
10226:^
10153:.
10131:^
10115:.
10030:.
9908:+
9870:+
9820:Re
9812:+
9752:).
9728::
9717:.
9711:−1
9237:.
9152:.
8690:+
8686:+
8682:+
8678:+
8649:−
8612:+
8608:+
8604:+
8550:+
7462:.
7339:φ(
7326:φ(
7314:,
7307:,
7105:.
6825:,
6657:j′
6465:.
6425::
6420:=
6342:SR
6292:SP
6247:SP
6209:SR
6119::
6113:P(
6105:R(
6097:P(
6085:R(
6076:.
6016:SP
5990:.
5953:,
5915:.
5853:1.
5792:SR
5761:.
5621::
5492:.
5458:.
5274:,
5267:,
5265:−1
5214:=
5186:,
5182:,
5154:=
5147:=
5129:,
5125:,
4793:−
4719:10
4667:+
4663:+
4659:+
4515:+
4420:.
4268:0.
4097:1/
3882:.
3200:2π
3195:=
2993:.
2483:.
2398:.
2379:.
2353:xy
2232:P(
2218:R(
2209:.
2104::
2098:P(
2086:R(
2078:P(
2064:R(
2057:+1
2010:,
1849:ka
1835:ka
1711:=
1357:=
1352:,
1340:,
1316:≤
1301:=
1284:=
1279:,
1267:,
1246:.
1218:=
1036:kn
1034:+
1030:=
1025:,
1013:=
860:1.
753:.
741:≤
710:.
692:.
644:1.
459:1.
293:1.
188:.
10876:)
10874:ς
10867:)
10865:ψ
10833:)
10830:S
10826:δ
10799:)
10797:ρ
10780:)
10778:φ
10717:e
10710:t
10703:v
10689:.
10659:.
10637:.
10618:.
10570:.
10557:.
10473:.
10469::
10461::
10416:.
10391:.
10363:.
10359::
10329:.
10323::
10293:.
10279::
10255:.
10220:.
10190:.
10163:.
10149::
10125:.
10111::
10086:.
10044:.
10015:.
9927:D
9924:(
9920:3
9911:z
9906:z
9899:n
9889:D
9886:(
9882:2
9873:z
9868:z
9861:n
9840:D
9837:(
9833:Z
9823:z
9815:z
9810:z
9799:n
9791:n
9783:n
9777:.
9769:D
9766:(
9758:n
9747:D
9743:(
9735:n
9722:n
9707:1
9701:n
9633:n
9629:k
9625:k
9621:n
9598:Q
9593:/
9589:)
9586:)
9583:n
9579:/
9575:i
9569:2
9566:(
9557:(
9553:Q
9530:.
9526:Z
9522:n
9518:/
9513:Z
9487:n
9472:Q
9467:/
9463:)
9460:)
9457:n
9453:/
9449:i
9443:2
9440:(
9431:(
9427:Q
9403:.
9399:Q
9388:n
9380:n
9362:.
9359:)
9356:)
9353:n
9349:/
9345:i
9339:2
9336:(
9327:(
9323:Q
9309:n
9295:,
9291:Q
9280:n
9227:n
9219:n
9212:n
9196:n
9185:n
9165:n
9161:n
9150:)
9148:p
9144:d
9139:)
9137:d
9135:(
9132:p
9127:d
9122:p
9115:n
9110:t
9104:t
9090:,
9085:t
9081:p
9072:2
9068:p
9064:+
9059:1
9055:p
9032:t
9028:p
9013:2
9009:p
9000:1
8996:p
8975:,
8970:t
8966:p
8957:2
8953:p
8947:1
8943:p
8939:=
8936:n
8918:n
8914:n
8908:n
8903:n
8899:n
8891:n
8882:Φ
8867:z
8861:z
8847:.
8842:k
8838:z
8832:1
8826:p
8821:0
8818:=
8815:k
8807:=
8801:1
8795:z
8790:1
8782:p
8778:z
8771:=
8768:)
8765:z
8762:(
8757:p
8742:p
8738:p
8730:p
8721:z
8717:z
8713:z
8709:z
8707:(
8705:8
8703:Φ
8696:z
8694:+
8692:z
8688:z
8684:z
8680:z
8676:z
8672:z
8668:z
8664:z
8662:(
8660:7
8658:Φ
8651:z
8647:z
8643:z
8639:z
8635:z
8631:z
8627:z
8625:(
8623:6
8621:Φ
8614:z
8610:z
8606:z
8602:z
8598:z
8594:z
8590:z
8588:(
8586:5
8584:Φ
8577:z
8573:z
8569:z
8565:z
8563:(
8561:4
8559:Φ
8552:z
8548:z
8544:z
8540:z
8536:z
8534:(
8532:3
8530:Φ
8523:z
8519:z
8515:z
8511:z
8509:(
8507:2
8505:Φ
8498:z
8494:z
8492:(
8490:1
8488:Φ
8476:μ
8458:,
8452:)
8447:d
8444:n
8439:(
8430:)
8426:1
8418:d
8414:z
8409:(
8402:n
8397:|
8392:d
8384:=
8379:)
8376:d
8373:(
8365:)
8361:1
8352:d
8349:n
8344:z
8339:(
8332:n
8327:|
8322:d
8314:=
8311:)
8308:z
8305:(
8300:n
8261:)
8258:1
8255:+
8250:4
8246:z
8242:(
8239:)
8236:1
8233:+
8228:2
8224:z
8220:(
8217:)
8214:1
8211:+
8208:z
8205:(
8202:)
8199:1
8193:z
8190:(
8187:=
8180:1
8172:8
8168:z
8160:)
8157:1
8154:+
8151:z
8148:+
8143:2
8139:z
8135:+
8130:3
8126:z
8122:+
8117:4
8113:z
8109:+
8104:5
8100:z
8096:+
8091:6
8087:z
8083:(
8080:)
8077:1
8071:z
8068:(
8065:=
8058:1
8050:7
8046:z
8038:)
8035:1
8032:+
8029:z
8021:2
8017:z
8013:(
8010:)
8007:1
8004:+
8001:z
7998:+
7993:2
7989:z
7985:(
7982:)
7979:1
7976:+
7973:z
7970:(
7967:)
7964:1
7958:z
7955:(
7952:=
7945:1
7937:6
7933:z
7925:)
7922:1
7919:+
7916:z
7913:+
7908:2
7904:z
7900:+
7895:3
7891:z
7887:+
7882:4
7878:z
7874:(
7871:)
7868:1
7862:z
7859:(
7856:=
7849:1
7841:5
7837:z
7829:)
7826:1
7823:+
7818:2
7814:z
7810:(
7807:)
7804:1
7801:+
7798:z
7795:(
7792:)
7789:1
7783:z
7780:(
7777:=
7770:1
7762:4
7758:z
7750:)
7747:1
7744:+
7741:z
7738:+
7733:2
7729:z
7725:(
7722:)
7719:1
7713:z
7710:(
7707:=
7700:1
7692:3
7688:z
7680:)
7677:1
7674:+
7671:z
7668:(
7665:)
7662:1
7656:z
7653:(
7650:=
7643:1
7635:2
7631:z
7623:1
7617:z
7614:=
7607:1
7599:1
7595:z
7575:z
7553:.
7550:)
7547:z
7544:(
7539:d
7529:n
7524:|
7519:d
7511:=
7508:1
7500:n
7496:z
7482:n
7478:d
7471:d
7467:n
7443:,
7437:1
7431:)
7428:1
7425:+
7422:z
7419:(
7414:1
7406:n
7402:)
7398:1
7395:+
7392:z
7389:(
7369:n
7361:)
7359:z
7357:(
7354:n
7351:Φ
7343:)
7341:n
7335:n
7330:)
7328:n
7323:z
7319:3
7316:z
7312:2
7309:z
7305:1
7302:z
7284:)
7279:k
7275:z
7268:z
7265:(
7260:)
7257:n
7254:(
7246:1
7243:=
7240:k
7232:=
7229:)
7226:z
7223:(
7218:n
7200:n
7187:n
7179:n
7162:1
7154:n
7150:z
7146:=
7143:)
7140:z
7137:(
7134:p
7103:)
7101:n
7097:n
7095:(
7093:O
7084:)
7082:n
7080:(
7078:O
7073:U
7061:U
7044:,
7035:j
7031:,
7028:j
7020:=
7011:j
7007:,
7004:k
7000:U
6986:k
6983:,
6980:j
6976:U
6968:n
6963:1
6960:=
6957:k
6935:U
6931:)
6929:n
6927:(
6924:O
6895:k
6889:j
6885:z
6874:2
6871:1
6862:n
6858:=
6853:k
6850:,
6847:j
6843:U
6829:)
6827:k
6823:j
6821:(
6817:U
6810:n
6806:n
6798:n
6794:z
6763:j
6759:,
6756:j
6745:n
6742:=
6737:k
6727:j
6722:z
6708:k
6702:j
6698:z
6690:n
6685:1
6682:=
6679:k
6661:n
6651:n
6647:j
6620:.
6615:s
6610:n
6607:a
6602:i
6596:2
6592:e
6586:n
6579:1
6576:=
6573:)
6570:n
6567:,
6564:a
6561:(
6553:1
6550:=
6547:a
6537:=
6534:)
6531:s
6528:(
6523:n
6519:c
6505:n
6501:s
6497:)
6495:s
6493:(
6490:n
6486:c
6475:n
6471:c
6463:)
6461:n
6459:(
6457:μ
6453:n
6441:d
6437:/
6433:n
6422:n
6418:d
6407:d
6403:/
6399:n
6388:n
6384:d
6366:.
6362:)
6357:d
6354:n
6349:(
6339:)
6336:d
6333:(
6325:n
6320:|
6315:d
6307:=
6304:)
6301:n
6298:(
6262:.
6259:)
6256:d
6253:(
6242:n
6237:|
6232:d
6224:=
6221:)
6218:n
6215:(
6183:,
6180:)
6177:d
6174:(
6168:P
6163:n
6158:|
6153:d
6145:=
6142:)
6139:n
6136:(
6130:R
6117:)
6115:n
6109:)
6107:n
6101:)
6099:n
6093:n
6089:)
6087:n
6070:)
6068:n
6066:(
6064:μ
6046:,
6043:)
6040:n
6037:(
6031:=
6028:)
6025:n
6022:(
6003:n
5999:)
5997:n
5986:n
5980:)
5978:n
5974:n
5970:z
5964:S
5960:S
5956:z
5947:n
5942:S
5935:z
5928:n
5921:n
5911:n
5904:n
5897:n
5886:X
5881:n
5847:n
5842:,
5839:0
5832:1
5829:=
5826:n
5821:,
5818:1
5812:{
5807:=
5804:)
5801:n
5798:(
5779:n
5775:)
5773:n
5742:.
5737:n
5733:k
5730:j
5724:2
5710:k
5706:B
5700:k
5692:+
5687:n
5683:k
5680:j
5674:2
5660:k
5656:A
5650:k
5642:=
5637:j
5633:x
5609:j
5605:x
5585:n
5578:2
5566:i
5563:+
5558:n
5551:2
5539:=
5533:n
5529:i
5523:2
5517:e
5513:=
5510:z
5497:n
5484:k
5480:X
5471:k
5467:j
5456:j
5450:n
5446:X
5442:1
5439:X
5419:j
5413:n
5409:z
5400:n
5396:X
5392:+
5386:+
5381:j
5375:1
5371:z
5365:1
5361:X
5357:=
5352:j
5346:k
5342:z
5333:k
5329:X
5323:k
5315:=
5310:j
5306:x
5292:n
5279:1
5276:x
5272:0
5269:x
5262:x
5253:n
5246:n
5232:n
5230:s
5226:1
5223:s
5221:{
5216:z
5212:z
5207:n
5202:n
5198:k
5188:z
5184:z
5180:z
5175:k
5173:s
5165:n
5161:j
5156:z
5152:z
5149:z
5145:z
5140:n
5131:z
5127:z
5123:z
5114:n
5110:z
5081:.
5076:2
5072:2
5066:i
5058:2
5054:2
5037:i
5029:x
5025:x
5023:(
5021:8
5019:Φ
5008:r
4994:,
4987:4
4982:2
4978:r
4969:1
4964:i
4956:2
4953:r
4931:,
4928:1
4922:r
4919:2
4911:2
4907:r
4903:+
4898:3
4894:r
4863:.
4858:2
4852:3
4847:i
4841:1
4832:,
4827:2
4821:3
4816:i
4813:+
4810:1
4795:x
4791:x
4787:x
4785:(
4783:6
4781:Φ
4744:,
4739:4
4733:5
4725:2
4722:+
4713:i
4705:4
4701:1
4693:5
4669:x
4665:x
4661:x
4657:x
4653:x
4651:(
4649:5
4647:Φ
4642:.
4639:i
4637:−
4632:i
4625:x
4621:x
4619:(
4617:4
4615:Φ
4599:.
4594:2
4588:3
4583:i
4577:1
4565:,
4560:2
4554:3
4549:i
4546:+
4543:1
4517:x
4513:x
4509:x
4507:(
4505:3
4503:Φ
4490:n
4484:n
4477:x
4473:x
4471:(
4469:2
4467:Φ
4460:n
4455:n
4448:x
4444:x
4442:(
4440:1
4438:Φ
4432:n
4410:n
4402:n
4386:n
4382:R
4359:.
4352:2
4347:)
4342:2
4339:r
4334:(
4326:1
4321:i
4298:,
4293:2
4290:r
4265:=
4262:1
4259:+
4256:z
4253:r
4245:2
4241:z
4215:n
4211:R
4200:n
4194:n
4191:Φ
4187:r
4171:n
4167:R
4150:n
4147:Φ
4143:z
4127:z
4124:1
4119:+
4116:z
4113:=
4110:r
4099:z
4093:n
4089:z
4073:n
4063:n
4049:n
4034:k
4030:k
4026:n
4022:n
4004:n
4000:1
3981:n
3974:n
3971:Φ
3966:n
3955:n
3952:Φ
3946:n
3943:Φ
3934:n
3928:n
3921:n
3906:x
3893:n
3866:)
3863:n
3859:/
3855:k
3849:2
3846:(
3822:n
3818:k
3809:n
3805:/
3801:k
3792:n
3775:.
3772:n
3766:k
3760:0
3756:,
3749:n
3746:k
3741:i
3735:2
3731:e
3717:n
3713:x
3696:,
3693:x
3684:i
3681:+
3678:x
3669:=
3664:x
3661:i
3657:e
3620:n
3613:n
3602:n
3596:n
3585:n
3566:n
3559:2
3547:i
3544:+
3539:n
3532:2
3508:n
3504:k
3486:1
3478:n
3471:k
3468:2
3456:i
3453:+
3448:n
3441:k
3438:2
3426:=
3421:k
3415:)
3409:n
3402:2
3390:i
3387:+
3382:n
3375:2
3362:(
3334:,
3331:1
3328:=
3322:2
3313:i
3310:+
3304:2
3295:=
3290:n
3284:)
3278:n
3271:2
3259:i
3256:+
3251:n
3244:2
3231:(
3216:n
3207:n
3203:/
3193:x
3175:.
3172:x
3169:n
3160:i
3157:+
3154:x
3151:n
3142:=
3137:n
3132:)
3128:x
3119:i
3116:+
3113:x
3103:(
3088:n
3084:x
3048:.
3045:)
3042:n
3038:/
3031:2
3028:(
3019:2
2971:.
2968:)
2962:(
2958:Q
2945:n
2916:)
2910:k
2895:(
2888:k
2858:)
2852:(
2848:Q
2823:)
2817:(
2813:Q
2801:k
2795:n
2780:)
2774:(
2770:Q
2740:k
2711:n
2705:k
2699:n
2693:ω
2687:k
2670:.
2666:Q
2641:)
2635:(
2631:Q
2619:n
2604:)
2598:(
2594:Q
2579:ω
2573:n
2567:ω
2561:n
2545:Q
2516:)
2510:(
2506:Q
2490:n
2464:n
2458:ω
2452:n
2446:ω
2440:n
2431:n
2425:n
2419:n
2414:n
2405:n
2376:n
2370:m
2360:k
2351:(
2345:x
2343:(
2337:y
2330:x
2298:.
2295:n
2292:=
2289:)
2286:d
2283:(
2275:n
2270:|
2265:d
2247:)
2245:n
2243:(
2241:φ
2236:)
2234:n
2227:n
2222:)
2220:n
2206:n
2201:1
2196:n
2186:d
2168:,
2165:)
2162:d
2159:(
2153:P
2148:n
2143:|
2138:d
2130:=
2127:)
2124:n
2121:(
2115:R
2102:)
2100:n
2090:)
2088:n
2082:)
2080:n
2073:n
2068:)
2066:n
2052:n
2042:φ
2036:n
2031:)
2029:n
2027:(
2025:φ
2019:n
2013:z
2003:n
1997:k
1979:.
1973:)
1970:n
1967:,
1964:k
1961:(
1954:n
1949:=
1943:)
1940:n
1937:,
1934:k
1931:(
1925:k
1920:n
1917:k
1911:=
1906:k
1902:)
1899:n
1896:,
1893:k
1890:(
1878:=
1875:a
1862:n
1858:k
1844:n
1840:k
1830:k
1826:n
1808:)
1805:n
1802:,
1799:k
1796:(
1770:,
1764:)
1761:n
1758:,
1755:k
1752:(
1745:n
1740:=
1737:a
1723:a
1718:z
1713:z
1709:w
1703:n
1697:z
1680:.
1676:)
1673:4
1666:(
1661:4
1655:2
1635:1
1632:=
1627:4
1623:z
1619:=
1614:2
1610:z
1597:z
1591:n
1584:n
1569:,
1564:b
1560:z
1556:=
1551:a
1547:z
1525:)
1522:n
1515:(
1510:b
1504:a
1493:z
1478:.
1474:)
1471:n
1464:(
1459:b
1453:a
1429:b
1425:z
1421:=
1416:a
1412:z
1400:n
1394:z
1385:n
1372:n
1366:n
1359:z
1355:z
1349:z
1343:z
1337:z
1331:z
1324:z
1318:n
1314:b
1310:a
1303:z
1299:z
1293:n
1286:z
1282:z
1276:z
1270:z
1264:z
1258:n
1252:z
1244:n
1240:a
1231:n
1227:r
1220:z
1216:z
1210:z
1204:z
1186:.
1181:b
1177:z
1173:=
1168:k
1164:1
1158:b
1154:z
1150:=
1145:k
1141:)
1135:n
1131:z
1127:(
1122:b
1118:z
1114:=
1109:n
1106:k
1102:z
1096:b
1092:z
1088:=
1083:n
1080:k
1077:+
1074:b
1070:z
1066:=
1061:a
1057:z
1042:k
1032:b
1028:a
1022:n
1015:z
1011:z
1006:)
1004:n
1000:b
996:a
990:n
984:z
966:.
957:z
951:=
946:1
940:n
936:z
932:=
927:1
920:z
916:=
911:z
908:1
892:n
882:n
857:=
852:k
848:1
844:=
839:k
835:)
829:n
825:z
821:(
818:=
813:n
810:k
806:z
802:=
797:n
793:)
787:k
783:z
779:(
765:n
759:n
749:z
743:n
739:a
733:a
727:z
721:n
707:n
686:n
682:k
678:n
670:n
661:n
638:n
635:,
629:,
626:3
623:,
620:2
617:,
614:1
611:=
608:m
600:1
592:m
588:z
577:1
574:=
569:n
565:z
551:m
547:m
535:n
523:n
512:F
502:F
496:F
490:F
484:F
453:n
450:,
444:,
441:1
438:,
435:0
432:=
429:k
425:,
420:n
413:k
410:2
398:i
395:+
390:n
383:k
380:2
368:=
364:)
359:n
355:i
349:k
346:2
340:(
323:n
311:n
290:=
285:n
281:z
267:z
263:n
257:n
244:r
240:n
224:n
220:n
217:n
174:n
123:)
117:(
112:)
108:(
98:·
91:·
84:·
77:·
50:.
20:)
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