84:
74:
53:
22:
2337:
Infinite products do not make sense (there is no way to define them) unless there is a notion of limit (e.g. metric spaces/topological spaces/topological groups etc.). In particular, groups and rings do not have a notion of limit, so infinite products do not make sense, and so there is no need to say
2194:
Somebody (not myself) has noted that there is a mistake in the 'Non-examples' section concerning quadratic integer rings and
Heegner numbers, however they noted this in parentheses in the article. I have added a clarification tag and am also mentioning it here on the talk page. If anybody in the know
2295:
Well, mathematically spending, I view the “nonzero ring” requirement is a key assumption; it is equivalent to saying the zero ideal is proper and “proper” is an important part of the definition of a prime ideal. Unfortunately (or fortunately), the leading paragraph is often quoted like by Google and
2225:
I've recently had some trouble with edit warring myself, so this is me trying to do a good deed and help out here. I agree with Joel's edit removing "non-zero". It's the lead of a different article and is just serving to remind people what an integral domain is; we don't need to get pedantic about
2310:
I have also rephrased the definition to avoid ambiguity that a product there might be infinite. (Saying “a product of some things” always leaves a possibility of infinite product). Mathematically, a good definition is “a commutative ring in which a product of a finite number of nonzero elements is
662:
Good point, and I guess that's the answer to my question, but why does the factorization have to be finite? I know that's the way the article is worded, and that's the standard definition in the literature (I just looked at Lang's book) but what's wrong with requiring just a bijection between the
498:
Actually, I believe it is unknown whether there there are infinitely many quadratic rings of algebraic integers which are UFDs. It is known that there are only infinitely many such rings for d negative. In fact, there are only finitely may such negative d for which the ring of integers is half
318:
Sorry, this post is about two months late, but I can answer for completeness of this section. In the integral domain article under the heading "Divisibility, prime and irreducible elements" you can find a detailed definition of what a prime element is. 'Prime numbers' and 'prime elements of an
412:, which was described as a non-UFD. It is a UFD. With this correction, however, the exposition is awkward. It claims that most factor rings are not UFD's and then gives an example of one which is a UFD. Perhaps someone should write up a nice (and correct) counterexample.
1124:
Assuming that my suspicion that there is no such definition of degree that works in the general case is correct, the proof can be fixed without too much trouble (and if my suspicion is incorrect, I think it would be a good idea to clarify what is meant by degree): If
677:
So you want unique factorization to mean that each element (up to associates) is uniquely determined by the powers of irreducibles that divide it? That's an interesting idea, and one I haven't seen before. (of course, it has no place on WP until someone publishes it)
2226:
edge cases. If you really have to point this out, "nontrivial" would probably be best. The latest "1≠0" (needs spaces anyway) is unfortunately probably worse since it will just look utterly baffling to someone who doesn't understand what the shorthand refers to. –
812:
by the following (not necessarily exhaustive) chain" suggests that there is only one class of rings properly between integral domains and principal ideal domains and that that is the class of UFDs. But that's quite untrue, consider for example the natural class of
487:
It's some time I have been out of school, but I heard there is only a finite number of rings of the form { a+b sqrt(d) | a,b integers} (real quadratic ring extensions I think is the term), which are also unique factorization domains. Is it true?
207:
Why is this significant? \sin \left( \pi \,z \right) =\sin \left( \pi \,z \right) significant? It seems to me when a zero gets in the product the whole this will become zero. The zero will happen when z and n are equal, 1-z^2/z^2.
499:
factorial--i.e. given two irreducible factorizations of any element, say x_1 x_2...x_n = y_1 y_2...y_m, then n=m (Carlitz's theorem states that any ring of algebraic integers is an HFD if and only if the class number is at most 2).
319:
integral domain' are not the same thing. You can think of prime elements as a generalization of the idea of prime numbers to any integral domain. Indeed, prime numbers are prime elements of the ring of integers.
596:
1 isn't an irreducible, it's a unit. The statement of factorization into irreducibles either explicitly excludes units (as in the article) or allows units as associates of the empty product of irreducibles (1).
2296:
such; thus, the accuracy is rather important: when
Knowledge is accused for an inaccuracy, that accusation often refers to the leading section. I think “nontrivial” can work here (will make that change). —-
535:
Added the following two properties of UFDs (both of which are well-known): Any UFD is integrally closed, and a domain R is a UFD if and only if every nonzero prime ideal of R has a nonzero prime element.
140:
465:
Let R be any commutative ring. Then R/(XY-ZW) is not a UFD. It is clear that X, Y, Z, and W are all irreducibles, so the element XY=ZW has two factorizations into irreducible elements.
799:
I suppose it's supposed to mean there are interesting classes of rings it could contain but doesn't. For example, you could have
Noetherian domains between integral domains and UFDs.
307:
I'm sorry but I still feel the redirection of UF to UFD without explaining the simple concept of UF (which the user was expecting when they clicked on/searched for UF) is an error.
1017:
468:
This proof is incorrect. (Though, the statement is true). X,Y,Z,W are irreducible in R, but one has to prove that X+R(XY-ZW) is irreducible in R/(XY-ZW), which is more difficult.
410:
951:
1402:
1309:
1216:
1119:
1873:
1734:
2060:
1981:
1583:
1484:
2081:
Every finite domain is a field, which the polynomial ring over a finite field clearly is not. The article could be confusing UFDs with unique factorization rings. ᛭
1046:
2028:
1949:
1920:
1830:
1783:
1763:
1691:
1650:
1630:
1551:
1531:
1449:
1329:
1236:
1143:
888:
868:
454:
745:
Is it correct, that the formal power series over a field (or even a PID) constitute a UFD? I mean, the holomorphic functions can be regarded as a subring of C
192:
I also reverted the statement about the uniqueness of the expression, since it is simply wrong that the p's are a permutation of the q's: 2*4 = (-2) * (-4).
2413:
130:
568:. Units in unique factorization domains can't be written as a product of irreducibles (except for 1, which could be considered as the empty product). --
2408:
2368:
An infinite product may make sense in a topological ring, for instant; so it’s usually a better writing style not to leave such an “ambiguity”. —-
2111:(2014, Oct 9): Request to add GCD Domains in the inclusion chain. Even a greedier request would be a digraph with nodes, each with a phrase - {I, I
649:
There's no factorisation into irreducibles here. The function sin has infinitely many pairwise non-associated irreducible factors (to wit (z-nπ)).
310:
Does your algebra page reasoning explain why prime elements is being used instead of prime numbers or why prime elements links to integral domain?
106:
2063:
431:
redirects here, but there's no mention in the article. Could someone who knows about this either add a definition or remove the redirect?
2259:
2235:
550:
170:
to have the value one (or more generally the multiplicative identity element) in order to reduce the need for case analysis in proofs.
756:
443:
629:
292:
97:
58:
1048:
is a unit, so it doesn't prove the reducibility of X, but without clarification on what is meant by the degree of the element of
704:
668:
639:
224:
33:
2144:
and whose morphisms are given by divisibility (a preorder). (0 does not uniquely divide itself, so one can also define
582:
But they can (trivially) be written as a product of the irreducible 1 and a unit, as is allowed in the other products.
175:
189:
I removed the non-example, because the previous one wasn't a ring, so there is no hope that it could ever be a UFD.
1121:, it's not clear whether this is a counterexample the the idea that the degree in mind has the desired properties.
700:
664:
635:
504:
However, Gauss conjectured that there are infinitely many quadratic rings of integers that are UFDs with d: -->
2067:
21:
554:
2255:
2231:
2180:
822:
760:
171:
163:
2373:
2343:
2316:
2301:
2278:
2200:
536:
506:
451:
440:
300:
39:
2339:
2274:
2216:
2196:
956:
83:
340:
800:
752:
718:
679:
650:
598:
587:
549:
Fields have irreducibles? How can you write 2 as a product of irreducibles in, say, the rationals? --
519:
515:
263:
Please note also that we don't use computer notation * for product, ^ for exponent. You need to write
212:
893:
472:
You're saying that this proof is lacking evidence at the statement "X,Y,Z,W are irreducible in R"?--
2101:
2086:
616:"The ring of functions in a fixed number of complex variables holomorphic at the origin is a UFD."
216:
105:
on
Knowledge. If you would like to participate, please visit the project page, where you can join
2251:
2227:
2176:
818:
311:
247:
220:
89:
73:
52:
1451:, then the degree of the product of any two elements is the sum of the degrees of the factors.
850:
The article asserts, without explanation, that the ring can be "grade by degree" such that, if
2369:
2312:
2311:
nonzero”. But I can use this nice definition only if I am writing a textbook on algebra... —-
2297:
2247:
2220:
1334:
1241:
1148:
1051:
789:
573:
1835:
1696:
489:
255:
890:
is a product of two factors of the form listed, but this is not obvious. For example, take
784:
The phrase "not necessarily exhaustive" is obscure to me. What is the intended meaning?
2033:
1954:
1556:
1457:
814:
714:
583:
416:
2175:
A domain with trivial unit group necessarily has characteristic 2, because –1 is a unit.
1022:
2377:
2347:
2320:
2305:
2282:
2263:
2239:
2204:
2184:
2105:
2090:
2071:
826:
803:
793:
764:
722:
708:
682:
672:
653:
643:
601:
591:
577:
558:
539:
522:
509:
492:
476:
326:
228:
179:
2097:
2082:
1404:
to be the minimal degree of all the elements in the corresponding equivalence class in
436:
428:
196:
1986:
1925:
1878:
1788:
1768:
1739:
1655:
1635:
1588:
1536:
1489:
1407:
1314:
1221:
1128:
873:
853:
2402:
545:"unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields"
473:
323:
236:
167:
246:
Why say prime elements rather than prime numbers? Are prime elements prime numbers?
2133:. Then the factorization into primes (put in order using ≤) is unique on the nose.
785:
619:
I suppose this is fairly straightforward, but do we have a reference with a proof?
569:
240:
195:
I also removed the repetion of the fact that in UFD's, irreducible implies prime.
699:
Well, I'm adding this as a counter-example, using (basically) your explaination.
291:
I am reverting your edits, since they really relate to something different, the
102:
1331:
is an integral domain, in this case, we can define the degree of an element of
79:
713:
Infinite factorizations are not well-defined without convergence properties.
632:, but I think the factorization might not be unique. Anybody know for sure?
628:
but I'm not sure if this is an example or a counter-example! We have the
439:
is a UFD. I'll add this to the article if someone else has not already.
164:
http://userweb.cs.utexas.edu/users/EWD/transcriptions/EWD09xx/EWD993.html
1311:
under the canonical homomorphism) and therefore is not a UFD. So assume
625:"The ring of holomorphic functions in a single complex variable..."
749:. Now, doesn't the same argument as with the sinus work somehow?
2129:
is 1 and assume a fixed total ordering ≤ on the set of primes in
286:
Also, please add comments to the bottom of the page, not the top.
278:
15:
2148:
to have the endomorphisms of 0 correspond to the elements of
808:
It's not quite right as it stands. To say that UFDs are "
746:
518:('R has class number 1' is equivalent to 'R is a UFD').
505:
0. I haven't heard of anyone resolving this conjecture.
254:
Because this is an algebra page, defining everything in
1218:
is not an integral domain either (the zero-divisors in
2036:
1989:
1957:
1928:
1881:
1838:
1791:
1771:
1742:
1699:
1658:
1638:
1591:
1559:
1539:
1492:
1460:
1410:
1337:
1317:
1244:
1224:
1151:
1131:
1054:
1025:
959:
896:
876:
856:
343:
101:, a collaborative effort to improve the coverage of
2195:could resolve this point, it would be appreciated.
2273:Agreed. Feel free to make the edit you suggested.
2054:
2022:
1975:
1943:
1914:
1867:
1824:
1777:
1757:
1728:
1685:
1644:
1624:
1577:
1545:
1525:
1478:
1443:
1396:
1323:
1303:
1230:
1210:
1137:
1113:
1040:
1011:
945:
882:
862:
404:
2030:, though there may be a simpler way to show that
2156:is no longer a preorder.) Isomorphic objects of
1454:This last sentence I can see is true given that
564:Fields don't have irreducibles. 2 is a unit of
2168:is skeletal if and only if the unit group of
8:
2096:Nevermind, misunderstood free generation. ᛭
19:
47:
2035:
1988:
1956:
1927:
1880:
1854:
1837:
1790:
1770:
1741:
1715:
1698:
1657:
1637:
1590:
1558:
1538:
1491:
1459:
1409:
1368:
1336:
1316:
1275:
1243:
1223:
1182:
1150:
1130:
1085:
1053:
1024:
994:
958:
934:
922:
907:
895:
875:
855:
387:
374:
362:
342:
2077:Polynomial rings over UFDs are not UFDs.
483:question about quadratic ring extensions
49:
1553:is an integral domain. I can see that
7:
2160:correspond to associate elements of
95:This article is within the scope of
2121:Take a unique factorization domain
239:link go to integral domain and not
38:It is of interest to the following
2140:whose objects are the elements of
1012:{\displaystyle X=(xX+1)(xX^{2}+X)}
14:
2414:Mid-priority mathematics articles
630:Weierstrass factorization theorem
405:{\displaystyle K/(Y^{2}-X^{2}+1)}
293:fundamental theorem of arithmetic
115:Knowledge:WikiProject Mathematics
2409:Start-Class mathematics articles
1145:is not an integral domain, then
118:Template:WikiProject Mathematics
82:
72:
51:
20:
2190:Mistake in Non-examples section
946:{\displaystyle R=F_{2}/(x^{2})}
450:And I see someone already did.
166:Dijkstra pleads for defining a
135:This article has been rated as
2136:In general, define a category
2106:22:22, 25 September 2012 (UTC)
2091:10:11, 23 September 2012 (UTC)
2017:
1993:
1938:
1932:
1909:
1885:
1862:
1845:
1819:
1795:
1752:
1746:
1723:
1706:
1680:
1662:
1619:
1595:
1520:
1496:
1438:
1414:
1391:
1373:
1365:
1341:
1298:
1280:
1272:
1248:
1205:
1187:
1179:
1155:
1108:
1090:
1082:
1058:
1006:
984:
981:
966:
940:
927:
919:
913:
612:Rings of holomorphic functions
399:
367:
359:
347:
1:
1238:are sent to zero-divisors in
839:/(XY-ZW) Counterexample": -->
663:irreducibles, finite or not?
109:and see a list of open tasks.
1486:is homogeneous and prime in
2125:such that the only unit in
827:20:49, 31 August 2008 (UTC)
804:23:26, 12 August 2008 (UTC)
723:23:32, 12 August 2008 (UTC)
602:23:33, 12 August 2008 (UTC)
592:23:31, 12 August 2008 (UTC)
540:00:50, 14 August 2007 (UTC)
510:00:21, 14 August 2007 (UTC)
2430:
2378:12:49, 17 April 2020 (UTC)
2348:12:07, 17 April 2020 (UTC)
2338:that products are finite.
2321:17:28, 16 April 2020 (UTC)
2306:17:12, 16 April 2020 (UTC)
2283:16:53, 16 April 2020 (UTC)
2264:16:50, 16 April 2020 (UTC)
2240:16:49, 16 April 2020 (UTC)
780:Not necessarily exhaustive
765:10:43, 6 August 2010 (UTC)
229:19:33, 24 March 2013 (UTC)
1397:{\displaystyle R/(XY-ZW)}
1304:{\displaystyle R/(XY-ZW)}
1211:{\displaystyle R/(XY-ZW)}
1114:{\displaystyle R/(XY-ZW)}
578:08:24, 5 March 2008 (UTC)
559:07:19, 5 March 2008 (UTC)
477:02:16, 29 June 2006 (UTC)
455:15:36, 5 April 2006 (UTC)
444:15:34, 5 April 2006 (UTC)
327:00:51, 27 June 2006 (UTC)
199:23:56 Nov 30, 2002 (UTC)
134:
67:
46:
2205:17:11, 14 May 2019 (UTC)
2185:23:57, 24 May 2015 (UTC)
2072:02:09, 31 May 2011 (UTC)
1868:{\displaystyle X-(ZW/Y)}
1729:{\displaystyle X-(ZW/Y)}
835:R/(XY-ZW) Counterexample
794:18:08, 6 July 2008 (UTC)
709:02:16, 31 May 2008 (UTC)
683:20:34, 30 May 2008 (UTC)
673:19:51, 30 May 2008 (UTC)
654:13:08, 30 May 2008 (UTC)
644:05:01, 30 May 2008 (UTC)
523:15:41, 21 May 2008 (UTC)
493:19:25, 7 July 2006 (UTC)
419:18:18, 26 Apr 2005 (UTC)
337:I corrected the example
303:18:18, 26 Apr 2005 (UTC)
185:Edit of 30 November 2002
180:19:49, 3 July 2010 (UTC)
141:project's priority scale
2246:Dammit, fixing ping to
1652:is the factor field of
622:I'm think about adding
168:product of zero factors
158:Product of zero factors
98:WikiProject Mathematics
2056:
2024:
1977:
1945:
1916:
1875:divides an element of
1869:
1826:
1779:
1759:
1730:
1687:
1646:
1626:
1579:
1547:
1527:
1480:
1445:
1398:
1325:
1305:
1232:
1212:
1139:
1115:
1042:
1013:
947:
884:
864:
514:It is unresolved. See
461:Fix this remark (TODO)
406:
314:27 April 02:51 Taipei
250:27 April 01:33 Taipei
28:This article is rated
2057:
2055:{\displaystyle XY-ZW}
2025:
1978:
1976:{\displaystyle XY-ZW}
1946:
1917:
1870:
1827:
1780:
1760:
1731:
1688:
1647:
1627:
1580:
1578:{\displaystyle XY-ZW}
1548:
1528:
1481:
1479:{\displaystyle XY-ZW}
1446:
1399:
1326:
1306:
1233:
1213:
1140:
1116:
1043:
1014:
948:
885:
865:
531:Added some properties
407:
2034:
1987:
1955:
1926:
1879:
1836:
1789:
1769:
1740:
1697:
1656:
1636:
1589:
1557:
1537:
1490:
1458:
1408:
1335:
1315:
1242:
1222:
1149:
1129:
1052:
1041:{\displaystyle xX+1}
1023:
957:
894:
874:
854:
701:Baccala@freesoft.org
665:Baccala@freesoft.org
636:Baccala@freesoft.org
516:class number problem
341:
121:mathematics articles
870:is reducible, then
2117:Trivial unit group
2052:
2020:
1973:
1941:
1912:
1865:
1822:
1775:
1755:
1726:
1683:
1642:
1622:
1575:
1543:
1523:
1476:
1441:
1394:
1321:
1301:
1228:
1208:
1135:
1111:
1038:
1009:
943:
880:
860:
402:
90:Mathematics portal
34:content assessment
2266:
2023:{\displaystyle R}
1944:{\displaystyle F}
1915:{\displaystyle R}
1825:{\displaystyle R}
1778:{\displaystyle Y}
1758:{\displaystyle F}
1686:{\displaystyle R}
1645:{\displaystyle F}
1625:{\displaystyle R}
1546:{\displaystyle R}
1533:, and given that
1526:{\displaystyle R}
1444:{\displaystyle R}
1324:{\displaystyle R}
1231:{\displaystyle R}
1138:{\displaystyle R}
883:{\displaystyle X}
863:{\displaystyle X}
755:comment added by
721:
590:
333:Fixing an example
277:for that, or use
232:
215:comment added by
172:HenningThielemann
155:
154:
151:
150:
147:
146:
2421:
2245:
2224:
2113:
2112:
2061:
2059:
2058:
2053:
2029:
2027:
2026:
2021:
1982:
1980:
1979:
1974:
1950:
1948:
1947:
1942:
1921:
1919:
1918:
1913:
1874:
1872:
1871:
1866:
1858:
1831:
1829:
1828:
1823:
1784:
1782:
1781:
1776:
1764:
1762:
1761:
1756:
1735:
1733:
1732:
1727:
1719:
1692:
1690:
1689:
1684:
1651:
1649:
1648:
1643:
1631:
1629:
1628:
1623:
1584:
1582:
1581:
1576:
1552:
1550:
1549:
1544:
1532:
1530:
1529:
1524:
1485:
1483:
1482:
1477:
1450:
1448:
1447:
1442:
1403:
1401:
1400:
1395:
1372:
1330:
1328:
1327:
1322:
1310:
1308:
1307:
1302:
1279:
1237:
1235:
1234:
1229:
1217:
1215:
1214:
1209:
1186:
1144:
1142:
1141:
1136:
1120:
1118:
1117:
1112:
1089:
1047:
1045:
1044:
1039:
1018:
1016:
1015:
1010:
999:
998:
952:
950:
949:
944:
939:
938:
926:
912:
911:
889:
887:
886:
881:
869:
867:
866:
861:
847:
846:
842:
815:Dedekind domains
767:
717:
586:
411:
409:
408:
403:
392:
391:
379:
378:
366:
301:Charles Matthews
256:commutative ring
231:
209:
123:
122:
119:
116:
113:
92:
87:
86:
76:
69:
68:
63:
55:
48:
31:
25:
24:
16:
2429:
2428:
2424:
2423:
2422:
2420:
2419:
2418:
2399:
2398:
2397:
2214:
2212:
2192:
2119:
2079:
2032:
2031:
1985:
1984:
1953:
1952:
1924:
1923:
1877:
1876:
1834:
1833:
1787:
1786:
1767:
1766:
1738:
1737:
1695:
1694:
1654:
1653:
1634:
1633:
1587:
1586:
1555:
1554:
1535:
1534:
1488:
1487:
1456:
1455:
1406:
1405:
1333:
1332:
1313:
1312:
1240:
1239:
1220:
1219:
1147:
1146:
1127:
1126:
1050:
1049:
1021:
1020:
990:
955:
954:
930:
903:
892:
891:
872:
871:
852:
851:
848:
844:
840:
838:
837:
782:
750:
715:Septentrionalis
614:
584:Septentrionalis
547:
533:
485:
463:
452:Shawn M. O'Hare
441:Shawn M. O'Hare
426:
383:
370:
339:
338:
335:
210:
205:
187:
160:
120:
117:
114:
111:
110:
88:
81:
61:
32:on Knowledge's
29:
12:
11:
5:
2427:
2425:
2417:
2416:
2411:
2401:
2400:
2393:
2391:
2389:
2388:
2387:
2386:
2385:
2384:
2383:
2382:
2381:
2380:
2357:
2356:
2355:
2354:
2353:
2352:
2351:
2350:
2328:
2327:
2326:
2325:
2324:
2323:
2308:
2288:
2287:
2286:
2285:
2268:
2267:
2211:
2208:
2191:
2188:
2118:
2115:
2109:
2108:
2078:
2075:
2064:67.188.193.108
2051:
2048:
2045:
2042:
2039:
2019:
2016:
2013:
2010:
2007:
2004:
2001:
1998:
1995:
1992:
1983:divides it in
1972:
1969:
1966:
1963:
1960:
1940:
1937:
1934:
1931:
1911:
1908:
1905:
1902:
1899:
1896:
1893:
1890:
1887:
1884:
1864:
1861:
1857:
1853:
1850:
1847:
1844:
1841:
1821:
1818:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1774:
1754:
1751:
1748:
1745:
1725:
1722:
1718:
1714:
1711:
1708:
1705:
1702:
1682:
1679:
1676:
1673:
1670:
1667:
1664:
1661:
1641:
1621:
1618:
1615:
1612:
1609:
1606:
1603:
1600:
1597:
1594:
1574:
1571:
1568:
1565:
1562:
1542:
1522:
1519:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1475:
1472:
1469:
1466:
1463:
1440:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1416:
1413:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1371:
1367:
1364:
1361:
1358:
1355:
1352:
1349:
1346:
1343:
1340:
1320:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1278:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1227:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1185:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1134:
1110:
1107:
1104:
1101:
1098:
1095:
1092:
1088:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1037:
1034:
1031:
1028:
1008:
1005:
1002:
997:
993:
989:
986:
983:
980:
977:
974:
971:
968:
965:
962:
942:
937:
933:
929:
925:
921:
918:
915:
910:
906:
902:
899:
879:
859:
836:
833:
832:
831:
830:
829:
781:
778:
777:
776:
775:
774:
773:
772:
771:
770:
769:
768:
734:
733:
732:
731:
730:
729:
728:
727:
726:
725:
690:
689:
688:
687:
686:
685:
657:
656:
613:
610:
609:
608:
607:
606:
605:
604:
546:
543:
537:192.236.44.130
532:
529:
528:
527:
526:
525:
507:192.236.44.130
501:
500:
484:
481:
480:
479:
462:
459:
458:
457:
447:
446:
437:factorial ring
429:Factorial ring
425:
424:Factorial ring
422:
421:
420:
401:
398:
395:
390:
386:
382:
377:
373:
369:
365:
361:
358:
355:
352:
349:
346:
334:
331:
330:
329:
320:
305:
304:
297:
296:
288:
287:
283:
282:
274:
273:
272:
271:
265:
264:
260:
259:
237:prime elements
204:
201:
186:
183:
159:
156:
153:
152:
149:
148:
145:
144:
133:
127:
126:
124:
107:the discussion
94:
93:
77:
65:
64:
56:
44:
43:
37:
26:
13:
10:
9:
6:
4:
3:
2:
2426:
2415:
2412:
2410:
2407:
2406:
2404:
2396:
2392:
2379:
2375:
2371:
2367:
2366:
2365:
2364:
2363:
2362:
2361:
2360:
2359:
2358:
2349:
2345:
2341:
2336:
2335:
2334:
2333:
2332:
2331:
2330:
2329:
2322:
2318:
2314:
2309:
2307:
2303:
2299:
2294:
2293:
2292:
2291:
2290:
2289:
2284:
2280:
2276:
2272:
2271:
2270:
2269:
2265:
2261:
2257:
2253:
2252:Deacon Vorbis
2249:
2244:
2243:
2242:
2241:
2237:
2233:
2229:
2228:Deacon Vorbis
2222:
2218:
2209:
2207:
2206:
2202:
2198:
2189:
2187:
2186:
2182:
2178:
2177:GeoffreyT2000
2173:
2171:
2167:
2163:
2159:
2155:
2151:
2147:
2143:
2139:
2134:
2132:
2128:
2124:
2116:
2114:
2107:
2103:
2099:
2095:
2094:
2093:
2092:
2088:
2084:
2076:
2074:
2073:
2069:
2065:
2049:
2046:
2043:
2040:
2037:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1990:
1970:
1967:
1964:
1961:
1958:
1935:
1929:
1906:
1903:
1900:
1897:
1894:
1891:
1888:
1882:
1859:
1855:
1851:
1848:
1842:
1839:
1816:
1813:
1810:
1807:
1804:
1801:
1798:
1792:
1772:
1749:
1743:
1720:
1716:
1712:
1709:
1703:
1700:
1677:
1674:
1671:
1668:
1665:
1659:
1639:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1592:
1572:
1569:
1566:
1563:
1560:
1540:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1493:
1473:
1470:
1467:
1464:
1461:
1452:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1411:
1388:
1385:
1382:
1379:
1376:
1369:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1338:
1318:
1295:
1292:
1289:
1286:
1283:
1276:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1245:
1225:
1202:
1199:
1196:
1193:
1190:
1183:
1176:
1173:
1170:
1167:
1164:
1161:
1158:
1152:
1132:
1122:
1105:
1102:
1099:
1096:
1093:
1086:
1079:
1076:
1073:
1070:
1067:
1064:
1061:
1055:
1035:
1032:
1029:
1026:
1003:
1000:
995:
991:
987:
978:
975:
972:
969:
963:
960:
935:
931:
923:
916:
908:
904:
900:
897:
877:
857:
843:
834:
828:
824:
820:
819:Richard Pinch
816:
811:
810:characterized
807:
806:
805:
802:
798:
797:
796:
795:
791:
787:
779:
766:
762:
758:
754:
748:
744:
743:
742:
741:
740:
739:
738:
737:
736:
735:
724:
720:
716:
712:
711:
710:
706:
702:
698:
697:
696:
695:
694:
693:
692:
691:
684:
681:
676:
675:
674:
670:
666:
661:
660:
659:
658:
655:
652:
648:
647:
646:
645:
641:
637:
633:
631:
626:
623:
620:
617:
611:
603:
600:
595:
594:
593:
589:
585:
581:
580:
579:
575:
571:
567:
563:
562:
561:
560:
556:
552:
551:68.161.152.76
544:
542:
541:
538:
530:
524:
521:
517:
513:
512:
511:
508:
503:
502:
497:
496:
495:
494:
491:
482:
478:
475:
471:
470:
469:
466:
460:
456:
453:
449:
448:
445:
442:
438:
434:
433:
432:
430:
423:
418:
415:
414:
413:
396:
393:
388:
384:
380:
375:
371:
363:
356:
353:
350:
344:
332:
328:
325:
321:
317:
316:
315:
313:
312:Cyclotronwiki
308:
302:
299:
298:
294:
290:
289:
285:
284:
280:
276:
275:
269:
268:
267:
266:
262:
261:
257:
253:
252:
251:
249:
248:Cyclotronwiki
244:
242:
238:
235:Why does the
233:
230:
226:
222:
218:
214:
202:
200:
198:
193:
190:
184:
182:
181:
177:
173:
169:
165:
157:
142:
138:
132:
129:
128:
125:
108:
104:
100:
99:
91:
85:
80:
78:
75:
71:
70:
66:
60:
57:
54:
50:
45:
41:
35:
27:
23:
18:
17:
2394:
2390:
2340:Joel Brennan
2275:Joel Brennan
2248:TakuyaMurata
2221:TakuyaMurata
2217:Joel Brennan
2213:
2197:Joel Brennan
2193:
2174:
2172:is trivial.
2169:
2165:
2161:
2157:
2153:
2149:
2145:
2141:
2137:
2135:
2130:
2126:
2122:
2120:
2110:
2080:
1785:is prime in
1736:is prime in
1585:is prime in
1453:
1123:
849:
809:
783:
757:84.137.53.18
634:
627:
624:
621:
618:
615:
565:
548:
534:
486:
467:
464:
427:
417:Gary Kennedy
336:
309:
306:
245:
241:prime number
234:
211:— Preceding
206:
194:
191:
188:
161:
137:Mid-priority
136:
96:
62:Mid‑priority
40:WikiProjects
1832:so that if
1019:. Granted,
751:—Preceding
490:Samohyl Jan
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
2403:Categories
2395:References
2062:is prime.
1632:since, if
801:Algebraist
719:PMAnderson
680:Algebraist
651:Algebraist
599:Algebraist
588:PMAnderson
520:Algebraist
2098:LokiClock
2083:LokiClock
203:Couple Qs
197:AxelBoldt
2210:Non-zero
2152:so that
753:unsigned
474:Rschwieb
324:Rschwieb
225:contribs
217:Tejolson
213:unsigned
1951:, then
1693:, then
953:, then
786:Plclark
570:Zundark
139:on the
2260:videos
2256:carbon
2236:videos
2232:carbon
2164:, and
1765:, and
258:terms.
36:scale.
270:2 × 5
2374:talk
2370:Taku
2344:talk
2317:talk
2313:Taku
2302:talk
2298:Taku
2279:talk
2219:and
2201:talk
2181:talk
2102:talk
2087:talk
2068:talk
841:edit
823:talk
790:talk
761:talk
705:talk
669:talk
640:talk
574:talk
555:talk
221:talk
176:talk
2250:. –
1922:in
817:.
279:TeX
162:In
131:Mid
2405::
2376:)
2346:)
2319:)
2304:)
2281:)
2262:)
2258:•
2238:)
2234:•
2203:)
2183:)
2104:)
2089:)
2070:)
2044:−
1965:−
1843:−
1704:−
1567:−
1468:−
1383:−
1290:−
1197:−
1100:−
825:)
792:)
763:)
707:)
671:)
642:)
576:)
557:)
488:--
435:A
381:−
322:--
243:?
227:)
223:•
178:)
2372:(
2342:(
2315:(
2300:(
2277:(
2254:(
2230:(
2223::
2215:@
2199:(
2179:(
2170:R
2166:C
2162:R
2158:C
2154:C
2150:R
2146:C
2142:R
2138:C
2131:R
2127:R
2123:R
2100:(
2085:(
2066:(
2050:W
2047:Z
2041:Y
2038:X
2018:]
2015:W
2012:,
2009:Z
2006:,
2003:Y
2000:,
1997:X
1994:[
1991:R
1971:W
1968:Z
1962:Y
1959:X
1939:]
1936:X
1933:[
1930:F
1910:]
1907:W
1904:,
1901:Z
1898:,
1895:Y
1892:,
1889:X
1886:[
1883:R
1863:)
1860:Y
1856:/
1852:W
1849:Z
1846:(
1840:X
1820:]
1817:W
1814:,
1811:Z
1808:,
1805:Y
1802:,
1799:X
1796:[
1793:R
1773:Y
1753:]
1750:X
1747:[
1744:F
1724:)
1721:Y
1717:/
1713:W
1710:Z
1707:(
1701:X
1681:]
1678:W
1675:,
1672:Z
1669:,
1666:Y
1663:[
1660:R
1640:F
1620:]
1617:W
1614:,
1611:Z
1608:,
1605:Y
1602:,
1599:X
1596:[
1593:R
1573:W
1570:Z
1564:Y
1561:X
1541:R
1521:]
1518:W
1515:,
1512:Z
1509:,
1506:Y
1503:,
1500:X
1497:[
1494:R
1474:W
1471:Z
1465:Y
1462:X
1439:]
1436:W
1433:,
1430:Z
1427:,
1424:Y
1421:,
1418:X
1415:[
1412:R
1392:)
1389:W
1386:Z
1380:Y
1377:X
1374:(
1370:/
1366:]
1363:W
1360:,
1357:Z
1354:,
1351:Y
1348:,
1345:X
1342:[
1339:R
1319:R
1299:)
1296:W
1293:Z
1287:Y
1284:X
1281:(
1277:/
1273:]
1270:W
1267:,
1264:Z
1261:,
1258:Y
1255:,
1252:X
1249:[
1246:R
1226:R
1206:)
1203:W
1200:Z
1194:Y
1191:X
1188:(
1184:/
1180:]
1177:W
1174:,
1171:Z
1168:,
1165:Y
1162:,
1159:X
1156:[
1153:R
1133:R
1109:)
1106:W
1103:Z
1097:Y
1094:X
1091:(
1087:/
1083:]
1080:W
1077:,
1074:Z
1071:,
1068:Y
1065:,
1062:X
1059:[
1056:R
1036:1
1033:+
1030:X
1027:x
1007:)
1004:X
1001:+
996:2
992:X
988:x
985:(
982:)
979:1
976:+
973:X
970:x
967:(
964:=
961:X
941:)
936:2
932:x
928:(
924:/
920:]
917:x
914:[
909:2
905:F
901:=
898:R
878:X
858:X
845:]
821:(
788:(
759:(
747:X
703:(
667:(
638:(
572:(
566:Q
553:(
400:)
397:1
394:+
389:2
385:X
376:2
372:Y
368:(
364:/
360:]
357:Y
354:,
351:X
348:[
345:K
295:.
281:.
219:(
174:(
143:.
42::
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
