Hilbert's Theorem 90

In abstract algebra, Hilbert's Theorem 90 (or Satz 90) refers to an important result on cyclic extensions of fields (or to one of its generalizations) that leads to Kummer theory. In its most basic form, it states that if L/K is a cyclic extension of fields with Galois group G =Gal(L/K) generated by an element s and if a is an element of L of relative norm 1, then there exists b in L such that

a = s(b)/b.

The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's famous Zahlbericht (Hilbert 1897, 1998), although it is originally due to Kummer (1855, p.213, 1861). Often a more general theorem due to Emmy Noether (1933) is given the name, stating that if L/K is a finite Galois extension of fields with Galois group G =Gal(L/K), then the first cohomology group is trivial:

H1(G, L×) = {1}