Hilbert's Theorem 90

Remarks

For part 2, Dedekind's lemma gives $\gamma \in L$ such that $\beta = \sum_{\tau \in G} f(\tau)\, \tau(\gamma)$ is nonzero. Then $\sigma(\beta) = \sum_\tau \sigma(f(\tau))\, \sigma\tau(\gamma) = f(\sigma)^{-1} \sum_\tau f(\sigma\tau)\, \sigma\tau(\gamma) = f(\sigma)^{-1} \beta$. For part 1, if $N(\alpha) = 1$ then $f(\sigma^i) = \alpha\, \sigma(\alpha) \cdots \sigma^{i-1}(\alpha)$ is well defined on the cyclic group and satisfies the condition of part 2; conversely $N(\beta / \sigma(\beta)) = N(\beta)/N(\sigma(\beta)) = 1$.

Example: take $L = \Q(i)$ and $K = \Q$, with $\sigma$ complex conjugation. The elements of norm $1$ are the $a + bi$ with $a^2 + b^2 = 1$, and the theorem writes each as $(m + ni)/(m - ni) = (m^2 - n^2 + 2mni)/(m^2 + n^2)$ with integers $m, n$ not both zero. This recovers the parametrization of rational points on the unit circle and of Pythagorean triples.

Part 1 needs a generator $\sigma$ of the Galois group, so it applies only to cyclic extensions; for a general finite Galois extension one uses part 2.

Going deeper

For a student

The cohomological form $H^1(\Gal(L/K), L^\times) = 1$ is the starting point of Galois cohomology. It is the main step in Kummer theory. When $K$ contains a primitive $n$-th root of unity, it shows that every cyclic extension of degree $n$ is obtained by adjoining an $n$-th root. There is also an additive version: in a cyclic extension, the elements of trace $0$ are exactly the $\beta - \sigma(\beta)$, which gives Artin–Schreier theory in characteristic $p$.