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/βs.
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 Lx is the multiplicative group of L, then the first cohomology group is trivial: - H1(G, Lx) = {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, Lx) = {1}
remains true.
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