Hilbert Theorem 90

Let $L/K$ be a finite Galois extension with Galois group $G=\mathrm{Gal}(L/K)$. The modern formulation of Hilbert’s Theorem 90 states that the first Galois cohomology group $H^1(G,L^{*})$ is 0.

The original statement of Hilbert’s Theorem 90 differs somewhat from the modern formulation given above, and is nowadays regarded as a corollary of the above fact. In its original form, Hilbert’s Theorem 90 says that if $G$ is cyclic with generator $\sigma$, then an element $x\in L$ has norm 1 if and only if

$$x=y/\sigma (y)$$

for some $y\in L$. Note that elements of the form $y/\sigma (y)$ are obviously contained within the kernel of the norm map; it is the converse that forms the content of the theorem.