Hilbert's Theorem 90

In abstract algebra, Hilbert's Theorem 90 (or Satz 90) is 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 an extension of fields with cyclic Galois group G = Gal(L/K) generated by an element ${\displaystyle \sigma ,}$ and if ${\displaystyle a}$ is an element of L of relative norm 1, that is

${\displaystyle N(a):=a\,\sigma (a)\,\sigma ^{2}(a)\cdots \sigma ^{n-1}(a)=1,}$

then there exists ${\displaystyle b}$ in L such that

${\displaystyle a=b/\sigma (b).}$

The theorem takes its name from the fact that it is the 90th theorem in David Hilbert's 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 arbitrary Galois group G = Gal(L/K), then the first cohomology group of G, with coefficients in the multiplicative group of L, is trivial:

${\displaystyle H^{1}(G,L^{\times })=\{1\}.}$