Field norm

Remarks

The norm is multiplicative: $N_{L/K}(\alpha\beta) = N_{L/K}(\alpha) N_{L/K}(\beta)$, and $N_{L/K}(c) = c^{[L:K]}$ for $c \in K$. It is transitive in towers: $N_{M/K} = N_{L/K} \circ N_{M/L}$. If the minimal polynomial of $\alpha$ over $K$ is $x^d + a_{d-1}x^{d-1} + \cdots + a_0$, then $N_{L/K}(\alpha) = \left((-1)^d a_0\right)^{[L:K(\alpha)]}$. If $L/K$ is separable and $\sigma_1, \ldots, \sigma_n$ are the embeddings of $L$ over $K$ into an algebraic closure, then $N_{L/K}(\alpha) = \sigma_1(\alpha) \cdots \sigma_n(\alpha)$.

Example: $L = \Q(i)$, $K = \Q$ and $\alpha = a + bi$ with $a, b \in \Q$. Check: in the basis $1, i$, the map $m_\alpha$ sends $1 \mapsto a + bi$ and $i \mapsto -b + ai$, so its matrix is $\begin{pmatrix} a & -b \\ b & a \end{pmatrix}$ and $N(\alpha) = a^2 + b^2$.

Non-example: the complex absolute value $a + bi \mapsto \sqrt{a^2 + b^2}$ is not the norm of $\Q(i)/\Q$, since it does not take values in $\Q$; for example $\abs{1 + i} = \sqrt{2}$.