Hilbert's Theorem 90

Hilbert's Theorem 90 is a result on cyclic extensions of fields that plays a foundational role in Kummer theory. In its classical form, it states that if L is a cyclic extension of a field K with Galois group generated by an element sigma, then any element of L whose relative norm down to K equals 1 can be written as b divided by sigma of b, for some element b of L. The theorem takes its name from its position as the ninetieth theorem in David Hilbert's Zahlbericht, though it was originally proved earlier by Ernst Kummer; a more general version proved by Emmy Noether extends the same statement to any finite Galois extension by showing that the first cohomology group of the Galois group with coefficients in the multiplicative group of L is trivial.