“Shafarevich–Weil theorem”的意思、由来-开放百科全书

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Shafarevich–Weil theorem

释义

Statement
References

In algebraic number theory, the Shafarevich–Weil theorem relates the fundamental class of a Galois extension of local or global fields to an extension of Galois groups. It was introduced by {{harvs|txt|last=Shafarevich|year=1946|authorlink=Igor Shafarevich}} for local fields and by {{harvs|txt|last=Weil|year=1951|authorlink=André Weil}} for global fields.

Statement

Suppose that F is a global field, K is a normal extension of F, and L is an abelian extension of K. Then the Galois group Gal(L/F) is an extension of the group Gal(K/F) by the abelian group Gal(L/K), and this extension corresponds to an element of the cohomology group H²(Gal(K/F), Gal(L/K)). On the other hand, class field theory gives a fundamental class in H²(Gal(K/F),I_K) and a reciprocity law map from I_K to Gal(L/K). The Shafarevich–Weil theorem states that the class of the extension Gal(L/F) is the image of the fundamental class under the homomorphism of cohomology groups induced by the reciprocity law map {{harv|Artin|Tate|2009|loc=p.246}}.

Shafarevich stated his theorem for local fields in terms of division algebras rather than the fundamental class {{harv|Weil|1967}}. In this case, with L the maximal abelian extension of K, the extension Gal(L/F) corresponds under the reciprocity map to the normalizer of K in a division algebra of degree [K:F] over F, and Shafarevich's theorem states that the Hasse invariant of this division algebra is 1/[K:F]. The relation to the previous version of the theorem is that division algebras correspond to elements of a second cohomology group (the Brauer group) and under this correspondence the division algebra with Hasse invariant 1/[K:F] corresponds to the fundamental class.

References

{{Citation | last1=Artin | first1=Emil | author1-link=Emil Artin | last2=Tate | first2=John | author2-link=John Tate | title=Class field theory | origyear=1952 | url=https://books.google.com/books?isbn=978-0-8218-4426-7 | publisher=AMS Chelsea Publishing, Providence, RI | isbn=978-0-8218-4426-7 | mr=0223335 | year=2009}}
{{citation|mr=0018170

|last=Shafarevich|first= I. R.
|title=On Galois groups of p-adic fields.
|journal=C. R. (Doklady) Acad. Sci. URSS (N.S.) |volume=53|year=1946|pages= 15–16}} Reprinted in his collected works, pages 4–5

{{Citation | last1=Weil | first1=André | author1-link = André Weil | title=Sur la theorie du corps de classes | year=1951 | journal=Journal of the Mathematical Society of Japan | issn=0025-5645 | volume=3 | pages=1–35|doi=10.2969/jmsj/00310001|mr=0044569}}, reprinted in volume I of his collected papers, {{ISBN|0-387-90330-5}}
{{citation|mr=0234930

|last=Weil|first= André
|title=Basic number theory.
|series=Die Grundlehren der mathematischen Wissenschaften|volume=144 |publisher=Springer-Verlag New York, Inc., New York |year=1967 |isbn= 3-540-58655-5 |chapter=Appendix III:Shafarevitch's theorem|pages=301–307}}{{DEFAULTSORT:Shafarevich-Weil theorem}}

1 : Theorems in algebraic number theory

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