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 Shafarevich (1946) for local fields and by Weil (1951) 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 (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*]. 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.