What’s Hilbert’s theorem 90?
Solution
Hilbert’s Theorem 90 is the statement \[ H^1(G,L^\times)=0 \] for a finite Galois extension \(L/K\) with Galois group \(G\).
In the cyclic case, say \(G=\langle\sigma\rangle\) of order \(n\), this is equivalent to the norm-one formulation: \[ N_{L/K}(a)=1 \quad\Longleftrightarrow\quad \exists\,b\in L^\times\text{ such that } a=\frac{b}{\sigma(b)}. \] Indeed, for a cyclic group the \(1\)-cocycle condition is determined by the value at \(\sigma\), and the condition that this value define a cocycle is exactly that its norm be \(1\); coboundaries have the form \(b/\sigma(b)\) (up to the opposite convention).
There is also an additive analogue: \[ H^1(G,L)=0. \] For a cyclic extension this says \[ \operatorname{Tr}_{L/K}(a)=0 \quad\Longleftrightarrow\quad a=b-\sigma(b) \] for some \(b\in L\) (again up to the sign convention for \(1-\sigma\)).