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 σ , {\displaystyle \sigma ,} and if a {\displaystyle a} is an element of L of relative norm 1, that is N ( a ) := a σ ( a ) σ 2 ( a ) ⋯ σ n − 1 ( a ) = 1 , {\displaystyle N(a):=a\,\sigma (a)\,\sigma ^{2}(a)\cdots \sigma ^{n-1}(a)=1,} then there exists b {\displaystyle b} in L such that a = b / σ ( b ) .