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.

On the other hand, class field theory gives a fundamental class in H2(Gal(K/F),IK) and a reciprocity law map from IK 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 (Artin & Tate 2009, p.246).

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