Shafarevich–Weil Theorem

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 for local fields and by 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 .

Shafarevich stated his theorem for local fields in terms of division algebras rather than the fundamental class . 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

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* Reprinted in his collected works, pages 4–5
* , reprinted in volume I of his collected papers,
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{{DEFAULTSORT:Shafarevich-Weil theorem
Theorems in algebraic number theory