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 is an extension of the group Gal by the abelian group Gal, and this extension corresponds to an element of the cohomology group H2, Gal). On the other hand, class field theory gives a fundamental class in H2 and a reciprocity law map from IK to Gal. The Shafarevich–Weil theorem states that the class of the extension Gal 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 corresponds under the reciprocity map to the normalizer of K in a division algebra of degree over F, and Shafarevich's theorem states that the Hasse invariant of this division algebra is 1/. The relation to the previous version of the theorem is that division algebras correspond to elements of a second cohomology group and under this correspondence the division algebra with Hasse invariant 1/ corresponds to the fundamental class.