2992:
2551:
2246:
2655:
1953:
3598:
For global fields there is a local-to-global principle: two global fields are Witt equivalent if and only if there is a bijection between their places such that the corresponding local fields are Witt equivalent. In particular, two number fields
3418:
is a related construction generated by isometry classes of nonsingular quadratic spaces with addition given by orthogonal sum and multiplication given by tensor product. Since two spaces that differ by a hyperbolic plane are not identified in
2371:
1284:
1126:
3352:
to the graded Witt ring. Milnor showed also that this homomorphism sends elements divisible by 2 to zero and that it is surjective. In the same paper he made a conjecture that this homomorphism is an isomorphism for all fields
2865:
2446:
266:
1959:
880:
142:
of forms. It is additively generated by the classes of one-dimensional forms. Although classes may contain spaces of different dimension, the parity of the dimension is constant across a class and so rk :
3226:
1510:
2853:
2562:
3401:
3168:
1643:
2785:
3342:
3116:
2759:
3068:
2288:
3296:
3025:
2815:
2703:
3270:
3250:
286:
1132:
2987:{\displaystyle \langle \langle a_{1},\ldots ,a_{n}\rangle \rangle =(\langle a_{1}\rangle -\langle 1\rangle )\cdots (\langle a_{n}\rangle -\langle 1\rangle )}
2546:{\displaystyle 0\rightarrow \mathbf {Z} \rightarrow W(\mathbf {Q} )\rightarrow \mathbf {Z} /2\oplus \bigoplus _{p\neq 2}W(\mathbf {F} _{p})\rightarrow 0\ }
962:
4243:
of the sphere spectrum, In: Axiomatic, Enriched and
Motivic Homotopy Theory, pp. 219–260, J.P.C. Greenlees (ed.), 2004 Kluwer Academic Publishers.
892:
Certain invariants of a quadratic form can be regarded as functions on Witt classes. We have seen that dimension mod 2 is a function on classes: the
459:
is real, then the nilpotent elements are precisely those of finite additive order, and these in turn are the forms all of whose signatures are zero;
2241:{\displaystyle (d_{1},e_{1},f_{1})\cdot (d_{2},e_{2},f_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2},^{1+e_{1}e_{2}}f_{1}^{e_{2}}f_{2}^{e_{1}})\ .}
216:
4266:
Perlis, R.; Szymiczek, K.; Conner, P.E.; Litherland, R. (1994). "Matching Witts with global fields". In Jacob, William B.; et al. (eds.).
771:
4532:
4498:
3173:
1552:
897:
105:
4637:
4570:
4456:
4422:
4380:
4279:
1444:
329:
3423:, the inverse for the addition needs to be introduced formally through the construction that was discovered by Grothendieck (see
893:
4448:
1563:. The invariants of a form over a number field are precisely the dimension, discriminant, all local Hasse invariants and the
4490:
4414:
4271:
1564:
494:
2820:
707:
2279:
2650:{\displaystyle W(\mathbf {Q} )\cong \mathbf {Z} \oplus \mathbf {Z} /2\oplus \bigoplus _{p\neq 2}W(\mathbf {F} _{p})\ }
108:, the non-degenerate two-dimensional symmetric bilinear form with a norm 0 vector. Each class is represented by the
3473:
3469:
3436:
623:
619:
482:
are the elements for which some signature is zero; otherwise, the zero-divisors are exactly the fundamental ideal.
394:
3768:
101:
3524:
The
Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 1 modulo 4 is isomorphic to
2263:) obtained by mapping a class to discriminant, rank mod 2, and the sequence of Hasse invariants. The kernel is
4677:
4125:
2706:
89:
81:
1948:{\displaystyle (d_{1},e_{1},f_{1})+(d_{2},e_{2},f_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2},f_{1}f_{2})}
301:
93:
62:
3513:
with trivial multiplication in the second component. The element (1, 0) corresponds to the quadratic form ⟨
3367:
3121:
354:
is of characteristic not equal to 2, so that we may identify symmetric bilinear forms and quadratic forms.
3756:
1330:
4177:
175:
139:
3580:
2764:
3539:
The
Grothendieck-Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 it is
3357:(of characteristic different from 2). This became known as the Milnor conjecture on quadratic forms.
2438:
664:
167:
4593:
4402:
3670:
3662:
3301:
3073:
2716:
418:
293:
203:
50:
4186:
3424:
3361:
3033:
2366:{\displaystyle W(\mathbf {Q} )\rightarrow W(\mathbf {R} )\oplus \prod _{p}W(\mathbf {Q} _{p})\ .}
542:
160:
113:
3403:, leading to increased understanding of the structure of quadratic forms over arbitrary fields.
3683:
The resulting groups (and generalizations thereof) are known as the even-dimensional symmetric
4633:
4566:
4528:
4494:
4452:
4418:
4406:
4376:
4275:
4142:
587:
471:
437:
370:
135:
3275:
4643:
4602:
4576:
4538:
4512:
4470:
4428:
4386:
4347:
4339:
4312:
4285:
4196:
4134:
3000:
2790:
574:
325:
171:
4508:
4466:
4256:-Algebraic topology over a field. Lecture Notes in Mathematics 2052, Springer Verlag, 2012.
4154:
2376:
We make this concrete, and compute the image, by using the "second residue homomorphism" W(
4647:
4629:
4606:
4580:
4562:
4554:
4542:
4516:
4504:
4474:
4462:
4432:
4390:
4351:
4316:
4289:
4150:
3745:-groups are not 4-periodic for all rings, hence they provide a less exact generalization.
3229:
2679:
449:
402:
138:
of non-degenerate symmetric bilinear forms, with the group operation corresponding to the
31:
1626:
are +1, that the value on acomplex places is +1 and that the product of all the terms in
1279:{\displaystyle (d_{1},e_{1})\cdot (d_{2},e_{2})=(d_{1}^{e_{2}}d_{2}^{e_{1}},e_{1}e_{2}).}
4398:
3752:
3666:
3448:
3255:
3235:
1560:
386:
271:
4330:
Czogała, A. (1999). "Higher degree tame
Hilbert-symbol equivalence of number fields".
733:
The Witt ring of a local field with maximal ideal of norm congruent to 3 modulo 4 is (
4671:
4440:
4368:
4167:
Orlov, Dmitry; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for
3572:
1301:) to this obtained by mapping a class to discriminant and rank mod 2. The kernel is
1121:{\displaystyle (d_{1},e_{1})+(d_{2},e_{2})=((-1)^{e_{1}e_{2}}d_{1}d_{2},e_{1}+e_{2})}
727:
562:
398:
343:), though the term "Witt ring" is often also used for a completely different ring of
312:
is the exponent of the torsion in the Witt group, if this is finite, or ∞ otherwise.
297:
65:
58:
4620:
Balmer, Paul (2005). "Witt groups". In
Friedlander, Eric M.; Grayson, D. R. (eds.).
3624:
3568:
1556:
1544:
901:
749:
649:
602:
475:
406:
211:
85:
3439:
if and only if this is an isomorphism. The hyperbolic spaces generate an ideal in
17:
4550:
4120:
3345:
3027:
2710:
1630:
in +1. Let be the sequence of
Hilbert symbols: it satisfies the conditions on
1350:
703:
358:
344:
42:
35:
4588:
4482:
4200:
1321:
The triple of discriminant, rank mod 2 and Hasse invariant defines a map from
1290:
711:
684:
680:
445:
179:
54:
4146:
4303:
Szymiczek, Kazimierz (1997). "Hilbert-symbol equivalence of number fields".
1362:
527:
109:
4661:
3741:) being the Witt group of (−1)-quadratic forms (skew-symmetric); symmetric
3452:
3643:. Some variations and extensions of this condition, such as "tame degree
261:{\displaystyle \langle \!\langle w\rangle \!\rangle =\langle 1,-w\rangle }
3684:
3656:
414:
3501:
The
Grothendieck-Witt ring of any finite field of odd characteristic is
3344:) to zero. This means that this mapping defines a homomorphism from the
875:{\displaystyle \mathbf {Z} _{8}/\langle 2s,2t,s^{2},t^{2},st-4\rangle .}
4525:
Algebra. Volume II: Fields with
Structure, Algebras and Advanced Topics
4343:
4138:
432:
is not formally real, the fundamental ideal is the only prime ideal of
4493:, vol. 211 (Revised third ed.), New York: Springer-Verlag,
4191:
3674:
2672:
be a field of characteristic not equal to 2. The powers of the ideal
3564:
Grothendieck-Witt ring and motivic stable homotopy groups of spheres
900:
is again a well-defined function on Witt classes with values in the
3221:{\displaystyle \langle \langle a_{1},\ldots ,a_{n}\rangle \rangle }
4591:(1936), "Theorie der quadratischen Formen in beliebigen Korpern",
4561:. Ergebnisse der Mathematik und ihrer Grenzgebiete. Vol. 73.
597:-th power of the fundamental ideal is additively generated by the
3360:
The conjecture was proved by Dmitry Orlov, Alexander Vishik and
3730:) being the Witt group of (1)-quadratic forms (symmetric), and
3575:
is isomorphic to the motivic stable homotopy group of spheres π
3451:
gives the
Grothendieck-Witt ring the additional structure of a
1622:, subject to the condition that all but finitely many terms of
4268:
Recent advances in real algebraic geometry and quadratic forms
3627:, preserving degree 2 Hilbert symbols. In this case the pair (
2434:
to be the 2-adic valuation of the discriminant, taken mod 2).
1505:{\displaystyle W(K)=W(k)\oplus \langle \pi \rangle \cdot W(k)}
1309:
may be regarded as classifying graded quadratic extensions of
4375:. Series in Pure Mathematics. Vol. 2. World Scientific.
4413:. University Lecture Series. Vol. 28. Providence, RI:
2270:
The symbol ring is a realisation of the Brauer-Wall group.
3607:
are Witt equivalent if and only if there is a bijection
3232:
and maps
Steinberg elements (elements such that for some
714:
congruent to 1 modulo 4 is isomorphic to the group ring (
332:
to define the ring product. This is sometimes called the
3647:
Hilbert symbol equivalence", have also been studied.
3370:
3304:
3278:
3258:
3238:
3176:
3124:
3076:
3036:
3003:
2868:
2823:
2793:
2767:
2719:
2682:
2565:
2449:
2291:
1962:
1646:
1447:
1135:
965:
774:
397:
with respective to the ordering. The Witt ring is a
350:
To discuss the structure of this ring we assume that
274:
219:
3661:
Witt groups can also be defined in the same way for
2676:
of forms of even dimension ("fundamental ideal") in
2848:{\displaystyle \langle a\rangle -\langle 1\rangle }
1618:a sequence of elements ±1 indexed by the places of
4373:A Survey of Trace Forms of Algebraic Number Fields
4123:(1970), "Algebraic K-theory and quadratic forms",
3395:
3336:
3290:
3264:
3244:
3220:
3162:
3110:
3062:
3019:
2986:
2847:
2809:
2779:
2753:
2697:
2649:
2545:
2365:
2251:Then there is a surjective ring homomorphism from
2240:
1947:
1637:We define addition and multiplication as follows:
1504:
1278:
1120:
874:
280:
260:
100:if one can be obtained from the other by adding a
233:
223:
4270:. Contemp. Math. Vol. 155. Providence, RI:
2817:considered as an element of the Witt ring. Then
4594:Journal für die reine und angewandte Mathematik
4451:. Vol. 67. American Mathematical Society.
1361:of characteristic not equal to 2. There is an
956:. Addition and multiplication are defined by:
357:The kernel of the rank mod 2 homomorphism is a
170:in the Witt group have order a power of 2; the
3030:in a 1970 paper proved that the mapping from
4411:Cohomological invariants in Galois cohomology
8:
4221:Garibaldi, Merkurjev & Serre (2003) p.63
4047:Garibaldi, Merkurjev & Serre (2003) p.64
3571:showed that the Grothendieck-Witt ring of a
3215:
3212:
3180:
3177:
2978:
2972:
2966:
2953:
2941:
2935:
2929:
2916:
2907:
2904:
2872:
2869:
2842:
2836:
2830:
2824:
2774:
2768:
2660:where the first component is the signature.
1484:
1478:
866:
807:
255:
240:
234:
230:
224:
220:
4664:in the Springer encyclopedia of mathematics
4445:Introduction to Quadratic Forms over Fields
2859:and correspondingly a product of the form
4190:
3989:
3987:
3985:
3983:
3910:
3908:
3837:Milnor & Husemoller (1973) p. 72
3816:Milnor & Husemoller (1973) p. 66
3804:Milnor & Husemoller (1973) p. 65
3786:Milnor & Husemoller (1973) p. 14
3372:
3371:
3369:
3364:in 1996 (published in 2007) for the case
3322:
3309:
3303:
3277:
3257:
3237:
3206:
3187:
3175:
3151:
3132:
3123:
3096:
3087:
3081:
3075:
3054:
3044:
3035:
3008:
3002:
2960:
2923:
2898:
2879:
2867:
2822:
2801:
2792:
2766:
2739:
2730:
2724:
2718:
2681:
2635:
2630:
2611:
2596:
2591:
2583:
2572:
2564:
2525:
2520:
2501:
2486:
2481:
2470:
2456:
2448:
2348:
2343:
2330:
2315:
2298:
2290:
2221:
2216:
2211:
2199:
2194:
2189:
2177:
2167:
2156:
2146:
2133:
2117:
2107:
2092:
2087:
2082:
2070:
2065:
2060:
2041:
2028:
2015:
1996:
1983:
1970:
1961:
1936:
1926:
1911:
1901:
1896:
1874:
1864:
1845:
1832:
1816:
1803:
1790:
1780:
1768:
1758:
1753:
1725:
1712:
1699:
1680:
1667:
1654:
1645:
1446:
1264:
1254:
1239:
1234:
1229:
1217:
1212:
1207:
1188:
1175:
1156:
1143:
1134:
1109:
1096:
1083:
1073:
1061:
1051:
1046:
1018:
1005:
986:
973:
964:
845:
832:
802:
781:
776:
773:
273:
218:
3964:
3962:
3960:
3958:
3956:
3854:
3852:
3755:, forming one of the three terms of the
4175:with applications to quadratic forms",
3779:
2556:which can be written as an isomorphism
541:and constitute the minimal prime ideal
489:is an ordered field with positive cone
405:if and only if there are finitely many
92:. We say that two spaces equipped with
3880:
3878:
3812:
3810:
2713:, that is the direct sum of quotients
3868:
3866:
3864:
3824:
3822:
3396:{\displaystyle {\textrm {char}}(k)=0}
3163:{\displaystyle (a_{1},\ldots ,a_{n})}
7:
3595:if their Witt rings are isomorphic.
3521:is not a square in the finite field.
3427:). There is a natural homomorphism
2709:and one may consider the associated
104:, that is, zero or more copies of a
3641:degree 2 Hilbert symbol equivalence
2282:implies that there is an injection
898:Hasse invariant of a quadratic form
4065:Conner & Perlis (1984) p.16-17
2407:) we obtain a group homomorphism ∂
1357:, uniformiser π and residue field
61:whose elements are represented by
25:
3697:) and even-dimensional quadratic
2780:{\displaystyle \langle a\rangle }
1380:) which lifts the diagonal form ⟨
505:defines a ring homomorphism from
330:tensor product of quadratic forms
30:For Witt groups in the theory of
3487:is isomorphic to the group ring
2631:
2592:
2584:
2573:
2521:
2482:
2471:
2457:
2344:
2316:
2299:
777:
4449:Graduate Studies in Mathematics
4074:Conner & Perlis (1984) p.18
4056:Conner & Perlis (1984) p.16
4020:Conner & Perlis (1984) p.12
3751:-groups are central objects in
3435:given by dimension: a field is
2664:Witt ring and Milnor's K-theory
497:holds for quadratic forms over
421:in the multiplicative group of
288:a non-zero sum of squares. If
3483:The Grothendieck-Witt ring of
3464:The Grothendieck-Witt ring of
3384:
3378:
3157:
3125:
3051:
3037:
2981:
2950:
2944:
2913:
2692:
2686:
2641:
2626:
2577:
2569:
2534:
2531:
2516:
2478:
2475:
2467:
2461:
2453:
2354:
2339:
2320:
2312:
2306:
2303:
2295:
2229:
2153:
2126:
2053:
2047:
2008:
2002:
1963:
1942:
1919:
1893:
1883:
1854:
1851:
1825:
1750:
1740:
1737:
1731:
1692:
1686:
1647:
1499:
1493:
1472:
1466:
1457:
1451:
1270:
1200:
1194:
1168:
1162:
1136:
1115:
1043:
1033:
1030:
1024:
998:
992:
966:
799:
787:
762:is of order 32 and is given by
436:and consists precisely of the
365:, of the Witt ring termed the
1:
4491:Graduate Texts in Mathematics
4415:American Mathematical Society
4332:Abh. Math. Sem. Univ. Hamburg
4272:American Mathematical Society
3719:-groups are 4-periodic, with
3498:is a cyclic group of order 2.
3337:{\displaystyle a_{i}+a_{j}=1}
3111:{\displaystyle I^{n}/I^{n+1}}
2754:{\displaystyle I^{n}/I^{n+1}}
1567:coming from real embeddings.
526:. These prime ideals are in
409:; that is, if the squares in
88:will be assumed to be finite-
4230:Lam (2005) p.36, Theorem 3.5
2394:). Composed with the map W(
1547:. For quadratic forms over
904:of the field of definition.
517:, with kernel a prime ideal
3063:{\displaystyle (k^{*})^{n}}
1535:Witt ring of a number field
896:is also well-defined. The
4694:
3654:
3591:Two fields are said to be
3474:quadratically closed field
3470:algebraically closed field
2274:Witt ring of the rationals
1341:Witt ring of a local field
624:quadratically closed field
620:algebraically closed field
495:Sylvester's law of inertia
29:
4371:; Perlis, Robert (1984).
4239:, On the motivic stable π
4201:10.4007/annals.2007.165.1
3769:Reduced height of a field
577:and the set of orderings
296:, then the Witt group is
210:; it is generated by the
102:metabolic quadratic space
4559:Symmetric Bilinear Forms
4126:Inventiones Mathematicae
3923:Lorenz (2008) p. 36
3902:Lorenz (2008) p. 33
3884:Lorenz (2008) p. 31
3872:Lorenz (2008) p. 35
3828:Lorenz (2008) p. 37
3795:Lorenz (2008) p. 30
3619:and a group isomorphism
2663:
1582:), as a set of triples (
328:structure, by using the
94:symmetric bilinear forms
3637:reciprocity equivalence
3291:{\displaystyle i\neq j}
2280:Hasse–Minkowski theorem
1293:ring homomorphism from
561:). The bijection is a
463:has Krull dimension 1.
166:The elements of finite
84:not equal to two. All
4523:Lorenz, Falko (2008).
3941:Lam (2005) pp. 277–280
3932:Lam (2005) p. 282
3914:Lam (2005) p. 280
3858:Lam (2005) p. 395
3846:Lam (2005) p. 260
3757:surgery exact sequence
3611:between the places of
3447:is the quotient. The
3413:Grothendieck-Witt ring
3407:Grothendieck-Witt ring
3397:
3338:
3292:
3266:
3246:
3222:
3164:
3112:
3064:
3021:
3020:{\displaystyle I^{n}.}
2988:
2849:
2811:
2810:{\displaystyle ax^{2}}
2787:be the quadratic form
2781:
2755:
2699:
2651:
2547:
2367:
2242:
1949:
1506:
1280:
1122:
924:), as a set of pairs (
912:We define a ring over
876:
565:between MinSpec
282:
262:
4212:Lam (2005) p. 28
4178:Annals of Mathematics
3968:Lam (2005) p. 34
3893:Lam (2005) p. 32
3669:, and more generally
3398:
3339:
3293:
3267:
3247:
3223:
3165:
3113:
3065:
3022:
2989:
2850:
2812:
2782:
2756:
2700:
2652:
2548:
2368:
2243:
1950:
1559:corresponding to the
1507:
1281:
1123:
908:Rank and discriminant
877:
283:
263:
140:orthogonal direct sum
126:is the abelian group
4632:. pp. 539–579.
4403:Merkurjev, Alexander
4305:Tatra Mt. Math. Publ
4274:. pp. 365–387.
4121:Milnor, John Willard
3663:skew-symmetric forms
3437:quadratically closed
3368:
3302:
3276:
3256:
3236:
3174:
3122:
3074:
3034:
3001:
2866:
2821:
2791:
2765:
2717:
2698:{\displaystyle W(k)}
2680:
2563:
2447:
2439:split exact sequence
2289:
1960:
1644:
1523:) with its image in
1445:
1133:
963:
772:
272:
217:
3625:square-class groups
2228:
2206:
2099:
2077:
1305:. The elements of
1246:
1224:
702:The Witt ring of a
648:The Witt ring of a
530:with the orderings
304:a power of 2. The
204:Pythagorean closure
136:equivalence classes
4407:Serre, Jean-Pierre
4344:10.1007/bf02940871
4139:10.1007/BF01425486
3615:and the places of
3581:A¹ homotopy theory
3443:and the Witt ring
3425:Grothendieck group
3393:
3362:Vladimir Voevodsky
3334:
3288:
3262:
3242:
3218:
3160:
3108:
3060:
3017:
2984:
2845:
2807:
2777:
2751:
2705:form a descending
2695:
2647:
2622:
2543:
2512:
2363:
2335:
2238:
2207:
2185:
2078:
2056:
1945:
1502:
1276:
1225:
1203:
1118:
872:
438:nilpotent elements
385:correspond to the
371:ring homomorphisms
320:The Witt group of
278:
258:
114:Witt decomposition
4534:978-0-387-72487-4
4500:978-0-387-95385-4
4369:Conner, Pierre E.
3715:). The quadratic
3671:ε-quadratic forms
3468:, and indeed any
3375:
3265:{\displaystyle j}
3245:{\displaystyle i}
2997:is an element of
2855:is an element of
2646:
2607:
2542:
2497:
2359:
2326:
2234:
1331:Brauer–Wall group
1317:Brauer–Wall group
755:The Witt ring of
637:The Witt ring of
618:, and indeed any
614:The Witt ring of
588:Harrison topology
472:Pythagorean field
367:fundamental ideal
281:{\displaystyle w}
18:Witt ring (forms)
16:(Redirected from
4685:
4651:
4609:
4584:
4555:Husemoller, Dale
4546:
4519:
4478:
4436:
4394:
4356:
4355:
4327:
4321:
4320:
4300:
4294:
4293:
4263:
4257:
4250:
4244:
4237:
4231:
4228:
4222:
4219:
4213:
4210:
4204:
4203:
4194:
4164:
4158:
4157:
4117:
4111:
4110:Lam (2005) p.178
4108:
4102:
4101:Lam (2005) p.175
4099:
4093:
4092:Lam (2005) p.174
4090:
4084:
4083:Lam (2005) p.116
4081:
4075:
4072:
4066:
4063:
4057:
4054:
4048:
4045:
4039:
4038:Lam (2005) p.117
4036:
4030:
4029:Lam (2005) p.113
4027:
4021:
4018:
4012:
4011:Lam (2005) p.119
4009:
4003:
4002:Lam (2005) p.166
4000:
3994:
3993:Lam (2005) p.152
3991:
3978:
3975:
3969:
3966:
3951:
3950:Lam (2005) p.316
3948:
3942:
3939:
3933:
3930:
3924:
3921:
3915:
3912:
3903:
3900:
3894:
3891:
3885:
3882:
3873:
3870:
3859:
3856:
3847:
3844:
3838:
3835:
3829:
3826:
3817:
3814:
3805:
3802:
3796:
3793:
3787:
3784:
3587:Witt equivalence
3402:
3400:
3399:
3394:
3377:
3376:
3373:
3343:
3341:
3340:
3335:
3327:
3326:
3314:
3313:
3297:
3295:
3294:
3289:
3271:
3269:
3268:
3263:
3251:
3249:
3248:
3243:
3227:
3225:
3224:
3219:
3211:
3210:
3192:
3191:
3169:
3167:
3166:
3161:
3156:
3155:
3137:
3136:
3117:
3115:
3114:
3109:
3107:
3106:
3091:
3086:
3085:
3069:
3067:
3066:
3061:
3059:
3058:
3049:
3048:
3026:
3024:
3023:
3018:
3013:
3012:
2993:
2991:
2990:
2985:
2965:
2964:
2928:
2927:
2903:
2902:
2884:
2883:
2854:
2852:
2851:
2846:
2816:
2814:
2813:
2808:
2806:
2805:
2786:
2784:
2783:
2778:
2760:
2758:
2757:
2752:
2750:
2749:
2734:
2729:
2728:
2704:
2702:
2701:
2696:
2656:
2654:
2653:
2648:
2644:
2640:
2639:
2634:
2621:
2600:
2595:
2587:
2576:
2552:
2550:
2549:
2544:
2540:
2530:
2529:
2524:
2511:
2490:
2485:
2474:
2460:
2372:
2370:
2369:
2364:
2357:
2353:
2352:
2347:
2334:
2319:
2302:
2247:
2245:
2244:
2239:
2232:
2227:
2226:
2225:
2215:
2205:
2204:
2203:
2193:
2184:
2183:
2182:
2181:
2172:
2171:
2151:
2150:
2138:
2137:
2122:
2121:
2112:
2111:
2098:
2097:
2096:
2086:
2076:
2075:
2074:
2064:
2046:
2045:
2033:
2032:
2020:
2019:
2001:
2000:
1988:
1987:
1975:
1974:
1954:
1952:
1951:
1946:
1941:
1940:
1931:
1930:
1918:
1917:
1916:
1915:
1906:
1905:
1879:
1878:
1869:
1868:
1850:
1849:
1837:
1836:
1821:
1820:
1808:
1807:
1795:
1794:
1785:
1784:
1775:
1774:
1773:
1772:
1763:
1762:
1730:
1729:
1717:
1716:
1704:
1703:
1685:
1684:
1672:
1671:
1659:
1658:
1511:
1509:
1508:
1503:
1289:Then there is a
1285:
1283:
1282:
1277:
1269:
1268:
1259:
1258:
1245:
1244:
1243:
1233:
1223:
1222:
1221:
1211:
1193:
1192:
1180:
1179:
1161:
1160:
1148:
1147:
1127:
1125:
1124:
1119:
1114:
1113:
1101:
1100:
1088:
1087:
1078:
1077:
1068:
1067:
1066:
1065:
1056:
1055:
1023:
1022:
1010:
1009:
991:
990:
978:
977:
881:
879:
878:
873:
850:
849:
837:
836:
806:
786:
785:
780:
575:Zariski topology
326:commutative ring
287:
285:
284:
279:
267:
265:
264:
259:
172:torsion subgroup
106:hyperbolic plane
68:over the field.
32:algebraic groups
21:
4693:
4692:
4688:
4687:
4686:
4684:
4683:
4682:
4678:Quadratic forms
4668:
4667:
4658:
4640:
4630:Springer-Verlag
4628:. Vol. 2.
4619:
4616:
4614:Further reading
4587:
4573:
4563:Springer-Verlag
4549:
4535:
4522:
4501:
4481:
4459:
4439:
4425:
4399:Garibaldi, Skip
4397:
4383:
4367:
4364:
4359:
4329:
4328:
4324:
4302:
4301:
4297:
4282:
4265:
4264:
4260:
4251:
4247:
4242:
4238:
4234:
4229:
4225:
4220:
4216:
4211:
4207:
4172:
4166:
4165:
4161:
4119:
4118:
4114:
4109:
4105:
4100:
4096:
4091:
4087:
4082:
4078:
4073:
4069:
4064:
4060:
4055:
4051:
4046:
4042:
4037:
4033:
4028:
4024:
4019:
4015:
4010:
4006:
4001:
3997:
3992:
3981:
3977:Lam (2005) p.37
3976:
3972:
3967:
3954:
3949:
3945:
3940:
3936:
3931:
3927:
3922:
3918:
3913:
3906:
3901:
3897:
3892:
3888:
3883:
3876:
3871:
3862:
3857:
3850:
3845:
3841:
3836:
3832:
3827:
3820:
3815:
3808:
3803:
3799:
3794:
3790:
3785:
3781:
3777:
3765:
3736:
3725:
3710:
3667:quadratic forms
3659:
3653:
3651:Generalizations
3593:Witt equivalent
3589:
3578:
3566:
3497:
3461:
3409:
3366:
3365:
3318:
3305:
3300:
3299:
3274:
3273:
3254:
3253:
3234:
3233:
3202:
3183:
3172:
3171:
3147:
3128:
3120:
3119:
3092:
3077:
3072:
3071:
3050:
3040:
3032:
3031:
3004:
2999:
2998:
2956:
2919:
2894:
2875:
2864:
2863:
2819:
2818:
2797:
2789:
2788:
2763:
2762:
2735:
2720:
2715:
2714:
2678:
2677:
2666:
2629:
2561:
2560:
2519:
2445:
2444:
2437:We then have a
2433:
2430:= 2 we define ∂
2425:
2412:
2406:
2393:
2384:
2342:
2287:
2286:
2276:
2217:
2195:
2173:
2163:
2152:
2142:
2129:
2113:
2103:
2088:
2066:
2037:
2024:
2011:
1992:
1979:
1966:
1958:
1957:
1932:
1922:
1907:
1897:
1892:
1870:
1860:
1841:
1828:
1812:
1799:
1786:
1776:
1764:
1754:
1749:
1721:
1708:
1695:
1676:
1663:
1650:
1642:
1641:
1561:Hilbert symbols
1553:Hasse invariant
1537:
1443:
1442:
1438:. This yields
1433:
1420:
1411:
1402:
1395:
1386:
1353:with valuation
1343:
1319:
1260:
1250:
1235:
1213:
1184:
1171:
1152:
1139:
1131:
1130:
1105:
1092:
1079:
1069:
1057:
1047:
1042:
1014:
1001:
982:
969:
961:
960:
910:
890:
841:
828:
775:
770:
769:
761:
747:
659:
611:
585:
535:
525:
450:Krull dimension
403:Noetherian ring
387:field orderings
324:can be given a
318:
270:
269:
215:
214:
74:
39:
28:
23:
22:
15:
12:
11:
5:
4691:
4689:
4681:
4680:
4670:
4669:
4666:
4665:
4657:
4656:External links
4654:
4653:
4652:
4638:
4615:
4612:
4611:
4610:
4585:
4571:
4547:
4533:
4520:
4499:
4479:
4457:
4441:Lam, Tsit-Yuen
4437:
4423:
4395:
4381:
4363:
4360:
4358:
4357:
4322:
4295:
4280:
4258:
4252:Fabien Morel,
4245:
4240:
4232:
4223:
4214:
4205:
4170:
4159:
4133:(4): 318–344,
4112:
4103:
4094:
4085:
4076:
4067:
4058:
4049:
4040:
4031:
4022:
4013:
4004:
3995:
3979:
3970:
3952:
3943:
3934:
3925:
3916:
3904:
3895:
3886:
3874:
3860:
3848:
3839:
3830:
3818:
3806:
3797:
3788:
3778:
3776:
3773:
3772:
3771:
3764:
3761:
3753:surgery theory
3734:
3723:
3705:
3655:Main article:
3652:
3649:
3635:) is called a
3623:between their
3588:
3585:
3576:
3565:
3562:
3561:
3560:
3537:
3522:
3499:
3495:
3481:
3460:
3457:
3449:exterior power
3408:
3405:
3392:
3389:
3386:
3383:
3380:
3333:
3330:
3325:
3321:
3317:
3312:
3308:
3287:
3284:
3281:
3261:
3241:
3217:
3214:
3209:
3205:
3201:
3198:
3195:
3190:
3186:
3182:
3179:
3159:
3154:
3150:
3146:
3143:
3140:
3135:
3131:
3127:
3105:
3102:
3099:
3095:
3090:
3084:
3080:
3057:
3053:
3047:
3043:
3039:
3016:
3011:
3007:
2995:
2994:
2983:
2980:
2977:
2974:
2971:
2968:
2963:
2959:
2955:
2952:
2949:
2946:
2943:
2940:
2937:
2934:
2931:
2926:
2922:
2918:
2915:
2912:
2909:
2906:
2901:
2897:
2893:
2890:
2887:
2882:
2878:
2874:
2871:
2844:
2841:
2838:
2835:
2832:
2829:
2826:
2804:
2800:
2796:
2776:
2773:
2770:
2748:
2745:
2742:
2738:
2733:
2727:
2723:
2694:
2691:
2688:
2685:
2665:
2662:
2658:
2657:
2643:
2638:
2633:
2628:
2625:
2620:
2617:
2614:
2610:
2606:
2603:
2599:
2594:
2590:
2586:
2582:
2579:
2575:
2571:
2568:
2554:
2553:
2539:
2536:
2533:
2528:
2523:
2518:
2515:
2510:
2507:
2504:
2500:
2496:
2493:
2489:
2484:
2480:
2477:
2473:
2469:
2466:
2463:
2459:
2455:
2452:
2431:
2421:
2408:
2402:
2389:
2380:
2374:
2373:
2362:
2356:
2351:
2346:
2341:
2338:
2333:
2329:
2325:
2322:
2318:
2314:
2311:
2308:
2305:
2301:
2297:
2294:
2275:
2272:
2249:
2248:
2237:
2231:
2224:
2220:
2214:
2210:
2202:
2198:
2192:
2188:
2180:
2176:
2170:
2166:
2162:
2159:
2155:
2149:
2145:
2141:
2136:
2132:
2128:
2125:
2120:
2116:
2110:
2106:
2102:
2095:
2091:
2085:
2081:
2073:
2069:
2063:
2059:
2055:
2052:
2049:
2044:
2040:
2036:
2031:
2027:
2023:
2018:
2014:
2010:
2007:
2004:
1999:
1995:
1991:
1986:
1982:
1978:
1973:
1969:
1965:
1955:
1944:
1939:
1935:
1929:
1925:
1921:
1914:
1910:
1904:
1900:
1895:
1891:
1888:
1885:
1882:
1877:
1873:
1867:
1863:
1859:
1856:
1853:
1848:
1844:
1840:
1835:
1831:
1827:
1824:
1819:
1815:
1811:
1806:
1802:
1798:
1793:
1789:
1783:
1779:
1771:
1767:
1761:
1757:
1752:
1748:
1745:
1742:
1739:
1736:
1733:
1728:
1724:
1720:
1715:
1711:
1707:
1702:
1698:
1694:
1691:
1688:
1683:
1679:
1675:
1670:
1666:
1662:
1657:
1653:
1649:
1594: ) with
1570:We define the
1536:
1533:
1513:
1512:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1459:
1456:
1453:
1450:
1429:
1416:
1407:
1400:
1391:
1384:
1349:be a complete
1342:
1339:
1318:
1315:
1287:
1286:
1275:
1272:
1267:
1263:
1257:
1253:
1249:
1242:
1238:
1232:
1228:
1220:
1216:
1210:
1206:
1202:
1199:
1196:
1191:
1187:
1183:
1178:
1174:
1170:
1167:
1164:
1159:
1155:
1151:
1146:
1142:
1138:
1128:
1117:
1112:
1108:
1104:
1099:
1095:
1091:
1086:
1082:
1076:
1072:
1064:
1060:
1054:
1050:
1045:
1041:
1038:
1035:
1032:
1029:
1026:
1021:
1017:
1013:
1008:
1004:
1000:
997:
994:
989:
985:
981:
976:
972:
968:
909:
906:
889:
886:
885:
884:
883:
882:
871:
868:
865:
862:
859:
856:
853:
848:
844:
840:
835:
831:
827:
824:
821:
818:
815:
812:
809:
805:
801:
798:
795:
792:
789:
784:
779:
764:
763:
759:
753:
745:
731:
700:
679:≡ 3 mod 4 and
655:
646:
635:
610:
607:
581:
545:MinSpec
533:
521:
407:square classes
317:
316:Ring structure
314:
277:
257:
254:
251:
248:
245:
242:
239:
236:
232:
229:
226:
222:
121:Witt group of
82:characteristic
73:
70:
66:bilinear forms
53:, named after
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4690:
4679:
4676:
4675:
4673:
4663:
4660:
4659:
4655:
4649:
4645:
4641:
4639:3-540-23019-X
4635:
4631:
4627:
4623:
4618:
4617:
4613:
4608:
4604:
4600:
4596:
4595:
4590:
4586:
4582:
4578:
4574:
4572:3-540-06009-X
4568:
4564:
4560:
4556:
4552:
4548:
4544:
4540:
4536:
4530:
4526:
4521:
4518:
4514:
4510:
4506:
4502:
4496:
4492:
4488:
4484:
4480:
4476:
4472:
4468:
4464:
4460:
4458:0-8218-1095-2
4454:
4450:
4446:
4442:
4438:
4434:
4430:
4426:
4424:0-8218-3287-5
4420:
4416:
4412:
4408:
4404:
4400:
4396:
4392:
4388:
4384:
4382:9971-966-05-0
4378:
4374:
4370:
4366:
4365:
4361:
4353:
4349:
4345:
4341:
4337:
4333:
4326:
4323:
4318:
4314:
4310:
4306:
4299:
4296:
4291:
4287:
4283:
4281:0-8218-5154-3
4277:
4273:
4269:
4262:
4259:
4255:
4249:
4246:
4236:
4233:
4227:
4224:
4218:
4215:
4209:
4206:
4202:
4198:
4193:
4188:
4184:
4180:
4179:
4174:
4163:
4160:
4156:
4152:
4148:
4144:
4140:
4136:
4132:
4128:
4127:
4122:
4116:
4113:
4107:
4104:
4098:
4095:
4089:
4086:
4080:
4077:
4071:
4068:
4062:
4059:
4053:
4050:
4044:
4041:
4035:
4032:
4026:
4023:
4017:
4014:
4008:
4005:
3999:
3996:
3990:
3988:
3986:
3984:
3980:
3974:
3971:
3965:
3963:
3961:
3959:
3957:
3953:
3947:
3944:
3938:
3935:
3929:
3926:
3920:
3917:
3911:
3909:
3905:
3899:
3896:
3890:
3887:
3881:
3879:
3875:
3869:
3867:
3865:
3861:
3855:
3853:
3849:
3843:
3840:
3834:
3831:
3825:
3823:
3819:
3813:
3811:
3807:
3801:
3798:
3792:
3789:
3783:
3780:
3774:
3770:
3767:
3766:
3762:
3760:
3758:
3754:
3750:
3746:
3744:
3740:
3733:
3729:
3722:
3718:
3714:
3709:
3704:
3700:
3696:
3692:
3689:
3687:
3681:
3679:
3676:
3672:
3668:
3664:
3658:
3650:
3648:
3646:
3642:
3638:
3634:
3630:
3626:
3622:
3618:
3614:
3610:
3606:
3602:
3596:
3594:
3586:
3584:
3582:
3574:
3573:perfect field
3570:
3563:
3558:
3554:
3550:
3546:
3544:
3538:
3535:
3531:
3527:
3523:
3520:
3516:
3512:
3508:
3504:
3500:
3494:
3490:
3486:
3482:
3479:
3475:
3471:
3467:
3463:
3462:
3458:
3456:
3454:
3450:
3446:
3442:
3438:
3434:
3430:
3426:
3422:
3417:
3414:
3406:
3404:
3390:
3387:
3381:
3363:
3358:
3356:
3351:
3347:
3331:
3328:
3323:
3319:
3315:
3310:
3306:
3285:
3282:
3279:
3259:
3239:
3231:
3207:
3203:
3199:
3196:
3193:
3188:
3184:
3152:
3148:
3144:
3141:
3138:
3133:
3129:
3103:
3100:
3097:
3093:
3088:
3082:
3078:
3055:
3045:
3041:
3029:
3014:
3009:
3005:
2975:
2969:
2961:
2957:
2947:
2938:
2932:
2924:
2920:
2910:
2899:
2895:
2891:
2888:
2885:
2880:
2876:
2862:
2861:
2860:
2858:
2839:
2833:
2827:
2802:
2798:
2794:
2771:
2746:
2743:
2740:
2736:
2731:
2725:
2721:
2712:
2708:
2689:
2683:
2675:
2671:
2661:
2636:
2623:
2618:
2615:
2612:
2608:
2604:
2601:
2597:
2588:
2580:
2566:
2559:
2558:
2557:
2537:
2526:
2513:
2508:
2505:
2502:
2498:
2494:
2491:
2487:
2464:
2450:
2443:
2442:
2441:
2440:
2435:
2429:
2424:
2420:
2416:
2411:
2405:
2401:
2397:
2392:
2388:
2383:
2379:
2360:
2349:
2336:
2331:
2327:
2323:
2309:
2292:
2285:
2284:
2283:
2281:
2273:
2271:
2268:
2266:
2262:
2258:
2254:
2235:
2222:
2218:
2212:
2208:
2200:
2196:
2190:
2186:
2178:
2174:
2168:
2164:
2160:
2157:
2147:
2143:
2139:
2134:
2130:
2123:
2118:
2114:
2108:
2104:
2100:
2093:
2089:
2083:
2079:
2071:
2067:
2061:
2057:
2050:
2042:
2038:
2034:
2029:
2025:
2021:
2016:
2012:
2005:
1997:
1993:
1989:
1984:
1980:
1976:
1971:
1967:
1956:
1937:
1933:
1927:
1923:
1912:
1908:
1902:
1898:
1889:
1886:
1880:
1875:
1871:
1865:
1861:
1857:
1846:
1842:
1838:
1833:
1829:
1822:
1817:
1813:
1809:
1804:
1800:
1796:
1791:
1787:
1781:
1777:
1769:
1765:
1759:
1755:
1746:
1743:
1734:
1726:
1722:
1718:
1713:
1709:
1705:
1700:
1696:
1689:
1681:
1677:
1673:
1668:
1664:
1660:
1655:
1651:
1640:
1639:
1638:
1635:
1634:just stated.
1633:
1629:
1625:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1585:
1581:
1577:
1573:
1568:
1566:
1562:
1558:
1555:±1 for every
1554:
1551:, there is a
1550:
1546:
1542:
1534:
1532:
1530:
1526:
1522:
1518:
1496:
1490:
1487:
1481:
1475:
1469:
1463:
1460:
1454:
1448:
1441:
1440:
1439:
1437:
1432:
1428:
1424:
1421:is a unit of
1419:
1415:
1410:
1406:
1399:
1394:
1390:
1383:
1379:
1375:
1371:
1367:
1364:
1360:
1356:
1352:
1348:
1340:
1338:
1336:
1332:
1328:
1324:
1316:
1314:
1312:
1308:
1304:
1300:
1296:
1292:
1273:
1265:
1261:
1255:
1251:
1247:
1240:
1236:
1230:
1226:
1218:
1214:
1208:
1204:
1197:
1189:
1185:
1181:
1176:
1172:
1165:
1157:
1153:
1149:
1144:
1140:
1129:
1110:
1106:
1102:
1097:
1093:
1089:
1084:
1080:
1074:
1070:
1062:
1058:
1052:
1048:
1039:
1036:
1027:
1019:
1015:
1011:
1006:
1002:
995:
987:
983:
979:
974:
970:
959:
958:
957:
955:
951:
947:
943:
939:
935:
931:
927:
923:
919:
915:
907:
905:
903:
899:
895:
887:
869:
863:
860:
857:
854:
851:
846:
842:
838:
833:
829:
825:
822:
819:
816:
813:
810:
803:
796:
793:
790:
782:
768:
767:
766:
765:
758:
754:
751:
744:
740:
736:
732:
729:
728:Klein 4-group
725:
721:
717:
713:
709:
708:maximal ideal
705:
701:
698:
694:
690:
686:
682:
678:
674:
670:
666:
663:
658:
654:
651:
647:
644:
640:
636:
633:
629:
625:
621:
617:
613:
612:
608:
606:
604:
603:Pfister forms
600:
596:
591:
589:
584:
580:
576:
572:
568:
564:
563:homeomorphism
560:
556:
552:
548:
544:
540:
536:
529:
524:
520:
516:
512:
508:
504:
500:
496:
492:
488:
483:
481:
477:
476:zero-divisors
473:
469:
464:
462:
458:
453:
451:
447:
443:
439:
435:
431:
426:
424:
420:
416:
412:
408:
404:
400:
399:Jacobson ring
396:
392:
388:
384:
380:
376:
372:
368:
364:
360:
355:
353:
348:
346:
342:
338:
335:
331:
327:
323:
315:
313:
311:
308:of the field
307:
303:
299:
295:
294:formally real
291:
275:
252:
249:
246:
243:
237:
227:
213:
212:Pfister forms
209:
205:
201:
197:
193:
189:
185:
181:
177:
173:
169:
164:
162:
158:
154:
150:
146:
141:
137:
133:
129:
125:
124:
117:
115:
111:
107:
103:
99:
95:
91:
87:
86:vector spaces
83:
79:
71:
69:
67:
64:
60:
59:abelian group
56:
52:
48:
44:
37:
33:
19:
4625:
4622:Handbook of
4621:
4601:(3): 31–44,
4598:
4592:
4558:
4551:Milnor, John
4527:. Springer.
4524:
4486:
4444:
4410:
4372:
4335:
4331:
4325:
4308:
4304:
4298:
4267:
4261:
4253:
4248:
4235:
4226:
4217:
4208:
4192:math/0101023
4182:
4176:
4168:
4162:
4130:
4124:
4115:
4106:
4097:
4088:
4079:
4070:
4061:
4052:
4043:
4034:
4025:
4016:
4007:
3998:
3973:
3946:
3937:
3928:
3919:
3898:
3889:
3842:
3833:
3800:
3791:
3782:
3748:
3747:
3742:
3738:
3731:
3727:
3720:
3716:
3712:
3707:
3702:
3698:
3694:
3690:
3685:
3682:
3677:
3660:
3644:
3640:
3636:
3632:
3628:
3620:
3616:
3612:
3608:
3604:
3600:
3597:
3592:
3590:
3569:Fabien Morel
3567:
3556:
3552:
3548:
3542:
3540:
3533:
3529:
3525:
3518:
3514:
3510:
3506:
3502:
3492:
3488:
3484:
3477:
3465:
3444:
3440:
3432:
3428:
3420:
3415:
3412:
3410:
3359:
3354:
3349:
2996:
2856:
2673:
2669:
2667:
2659:
2555:
2436:
2427:
2422:
2418:
2414:
2409:
2403:
2399:
2395:
2390:
2386:
2381:
2377:
2375:
2277:
2269:
2264:
2260:
2256:
2252:
2250:
1636:
1631:
1627:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1569:
1557:finite place
1548:
1545:number field
1540:
1538:
1528:
1524:
1520:
1516:
1515:identifying
1514:
1435:
1430:
1426:
1422:
1417:
1413:
1408:
1404:
1397:
1392:
1388:
1381:
1377:
1373:
1369:
1365:
1358:
1354:
1346:
1344:
1334:
1326:
1322:
1320:
1310:
1306:
1302:
1298:
1294:
1288:
953:
949:
945:
941:
937:
933:
929:
925:
921:
917:
913:
911:
902:Brauer group
894:discriminant
891:
756:
750:cyclic group
742:
738:
734:
723:
719:
715:
696:
692:
688:
676:
672:
668:
661:
656:
652:
650:finite field
642:
638:
631:
627:
615:
598:
594:
592:
582:
578:
570:
566:
558:
554:
550:
546:
538:
531:
522:
518:
514:
510:
506:
502:
498:
490:
486:
484:
479:
467:
465:
460:
456:
454:
441:
433:
429:
427:
422:
410:
393:, by taking
390:
382:
378:
374:
366:
362:
356:
351:
349:
345:Witt vectors
340:
336:
333:
321:
319:
309:
305:
289:
207:
199:
195:
191:
187:
183:
165:
161:homomorphism
156:
152:
148:
144:
131:
127:
122:
120:
118:
97:
77:
76:Fix a field
75:
46:
40:
27:Algebra term
4589:Witt, Ernst
4483:Lang, Serge
4338:: 175–185.
4185:(1): 1–13,
3673:, over any
3346:Milnor ring
3230:multilinear
3118:that sends
3028:John Milnor
2711:graded ring
1572:symbol ring
1425:with image
1351:local field
752:of order 2.
704:local field
573:) with the
401:. It is a
359:prime ideal
90:dimensional
43:mathematics
36:Witt vector
4662:Witt rings
4648:1115.19004
4607:0015.05701
4581:0292.10016
4543:1130.12001
4517:0984.00001
4475:1068.11023
4433:1159.12311
4391:0551.10017
4362:References
4352:0968.11038
4317:0978.11012
4290:0807.11024
3665:, and for
3579:(S) (see "
3272:such that
2707:filtration
1565:signatures
1291:surjective
888:Invariants
699:≡ 1 mod 4.
685:group ring
681:isomorphic
470:is a real
446:local ring
417:of finite
180:functorial
98:equivalent
72:Definition
55:Ernst Witt
47:Witt group
4147:0020-9910
3701:-groups
3283:≠
3216:⟩
3213:⟩
3197:…
3181:⟨
3178:⟨
3142:…
3046:∗
2979:⟩
2973:⟨
2970:−
2967:⟩
2954:⟨
2948:⋯
2942:⟩
2936:⟨
2933:−
2930:⟩
2917:⟨
2908:⟩
2905:⟩
2889:…
2873:⟨
2870:⟨
2843:⟩
2837:⟨
2834:−
2831:⟩
2825:⟨
2775:⟩
2769:⟨
2616:≠
2609:⨁
2605:⊕
2589:⊕
2581:≅
2535:→
2506:≠
2499:⨁
2495:⊕
2479:→
2462:→
2454:→
2328:∏
2324:⊕
2307:→
2259:) to Sym(
2006:⋅
1887:−
1858:−
1744:−
1488:⋅
1485:⟩
1482:π
1479:⟨
1476:⊕
1363:injection
1329:) to the
1166:⋅
1037:−
867:⟩
861:−
808:⟨
586:with the
528:bijection
503:signature
474:then the
395:signature
334:Witt ring
256:⟩
250:−
241:⟨
235:⟩
231:⟩
225:⟨
221:⟨
198:), where
182:map from
110:core form
63:symmetric
4672:Category
4557:(1973).
4485:(2002),
4443:(2005).
4409:(2003).
4311:: 7–16.
3763:See also
3657:L-theory
3517:⟩ where
3491:, where
3459:Examples
3298:one has
1412:⟩ where
928:,
741:) where
722:) where
609:Examples
543:spectrum
501:and the
448:and has
415:subgroup
302:exponent
57:, is an
4626:-theory
4509:1878556
4487:Algebra
4467:2104929
4155:0260844
3688:-groups
2426:) (for
1614:/2 and
932:) with
726:is the
683:to the
413:form a
369:. The
300:, with
298:torsion
292:is not
202:is the
178:of the
174:is the
4646:
4636:
4605:
4579:
4569:
4541:
4531:
4515:
4507:
4497:
4473:
4465:
4455:
4431:
4421:
4389:
4379:
4350:
4315:
4288:
4278:
4153:
4145:
3675:*-ring
3453:λ-ring
2761:. Let
2645:
2541:
2417:) → W(
2398:) → W(
2385:) → W(
2358:
2233:
1578:, Sym(
1396:⟩ to ⟨
601:-fold
306:height
176:kernel
34:, see
4187:arXiv
3775:Notes
3639:or a
3476:, is
1574:over
1543:be a
748:is a
706:with
695:) if
660:with
626:, is
553:) of
513:) to
493:then
444:is a
419:index
381:) to
373:from
268:with
190:) to
168:order
159:is a
134:) of
112:of a
51:field
49:of a
4634:ISBN
4567:ISBN
4529:ISBN
4495:ISBN
4453:ISBN
4419:ISBN
4377:ISBN
4276:ISBN
4143:ISSN
3603:and
3583:").
3411:The
3374:char
3252:and
2668:Let
2413:: W(
2278:The
1539:Let
1403:,...
1387:,...
1372:) →
1345:Let
944:and
712:norm
593:The
151:) →
119:The
96:are
45:, a
4644:Zbl
4603:Zbl
4599:176
4577:Zbl
4539:Zbl
4513:Zbl
4471:Zbl
4429:Zbl
4387:Zbl
4348:Zbl
4340:doi
4313:Zbl
4286:Zbl
4197:doi
4183:165
4135:doi
3577:0,0
3528:⊕ (
3472:or
3348:of
3228:is
3170:to
3070:to
1610:in
1598:in
1531:).
1434:in
1337:).
1333:BW(
948:in
936:in
710:of
675:if
667:is
665:odd
641:is
622:or
537:of
485:If
478:of
466:If
455:If
452:0.
428:If
389:of
206:of
80:of
41:In
4674::
4642:.
4597:,
4575:.
4565:.
4553:;
4537:.
4511:,
4505:MR
4503:,
4489:,
4469:.
4463:MR
4461:.
4447:.
4427:.
4417:.
4405:;
4401:;
4385:.
4346:.
4336:69
4334:.
4309:11
4307:.
4284:.
4195:,
4181:,
4173:/2
4151:MR
4149:,
4141:,
4129:,
3982:^
3955:^
3907:^
3877:^
3863:^
3851:^
3821:^
3809:^
3759:.
3680:.
3631:,
3555:/2
3551:⊕
3547:/4
3543:⊕
3541:Z'
3536:).
3532:/2
3509:/2
3505:⊕
3455:.
3441:GW
3431:→
3429:GW
3421:GW
3416:GW
2267:.
1606:,
1604:K*
1600:K*
1590:,
1586:,
1313:.
952:/2
942:K*
938:K*
916:,
737:/4
718:/2
691:/2
671:/4
630:/2
605:.
590:.
440:;
425:.
361:,
347:.
163:.
155:/2
116:.
4650:.
4624:K
4583:.
4545:.
4477:.
4435:.
4393:.
4354:.
4342::
4319:.
4292:.
4254:A
4241:0
4199::
4189::
4171:*
4169:K
4137::
4131:9
3749:L
3743:L
3739:R
3737:(
3735:2
3732:L
3728:R
3726:(
3724:0
3721:L
3717:L
3713:R
3711:(
3708:k
3706:2
3703:L
3699:L
3695:R
3693:(
3691:L
3686:L
3678:R
3645:l
3633:t
3629:T
3621:t
3617:L
3613:K
3609:T
3605:L
3601:K
3559:.
3557:Z
3553:Z
3549:Z
3545:Z
3534:Z
3530:Z
3526:Z
3519:a
3515:a
3511:Z
3507:Z
3503:Z
3496:2
3493:C
3489:Z
3485:R
3480:.
3478:Z
3466:C
3445:W
3433:Z
3391:0
3388:=
3385:)
3382:k
3379:(
3355:k
3350:k
3332:1
3329:=
3324:j
3320:a
3316:+
3311:i
3307:a
3286:j
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3240:i
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3089:/
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3015:.
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2911:=
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2030:2
2026:e
2022:,
2017:2
2013:d
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2003:)
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1985:1
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1977:,
1972:1
1968:d
1964:(
1943:)
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1934:f
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