Hilbert's Theorem 90

In number theory, Hilbert's Theorem 90 tells us that if L/K is a cyclic extension of number fields generated by an element s and if α is an element of L of relative norm 1, then then there exists β in L such that

α = β/β^s.

The theorem has its most natural statement in terms of group cohomology, where if G is the Galois group

Gal(L/K)

of L over K, and L^x is the multiplicative group of L, then the first cohomology group is trivial:

H¹(G, L^x) = {1}.

The theorem takes its name from the fact that it is the 90th theorem in Hilbert's famous Zahlbericht of 1897. Often a more general theorem is given the name, stating that if L/K is a finite Galois extension of fields, then the first cohomology group is trivial;

H¹(G, L^x) = {1}

remains true.