Theorem 3 (*Hilbert*)

— *Let E be a cyclic extension of K and let a be a generator of the Galois group $\Gamma$ of E over K.

a) For an element $x \in E$ to satisfy $N_{E/K}(x) = 1$ it is necessary and sufficient that $y \in E^*$ exist such that $x = y / \sigma(y)$; every $y_1 \in E^*$ with $x = y_1 / \sigma(y_1)$ is then of the form $\lambda y$ where $\lambda \in K^*$.

b) For an element $x \in E$ to satisfy $\mathrm{Tr}_{E/K}(x) = 0$ it is necessary and sufficient that $z \in E$ exist with $x = z - \sigma(z)$; then every $z_1 \in E$ with $x = z_1 - \sigma(z_1)$ is of the form $z + \mu$, where $\mu \in K$.

Let us first prove a lemma.*