Field norm

Definition. Let $L/K$ be a finite field extension with $n=[L:K]$. For $\alpha\in L$, let $m_\alpha(x)=\alpha x$ be multiplication by $\alpha$. The norm of $\alpha$ from $L$ to $K$ is

the usual determinant of this $K$-linear operator.

The norm is multiplicative: $\mathrm{N}(\alpha\beta)=\mathrm{N}(\alpha)\mathrm{N}(\beta)$, and is compatible with towers (see norm in towers ).

See also. field trace , units (the norm sends $L^\times\to K^\times$).

Examples.

1. In $L=\mathbb{Q}(\sqrt d)$ (char $\neq 2$), $\mathrm{N}_{L/\mathbb{Q}}(a+b\sqrt d)=a^2-d b^2$.
2. In $\mathbb{C}/\mathbb{R}$, $\mathrm{N}_{\mathbb{C}/\mathbb{R}}(a+bi)=a^2+b^2$.
3. For $L=\mathbb{F}_{q^n}$ over $\mathbb{F}_q$, $\mathrm{N}_{L/\mathbb{F}_q}(\alpha)=\alpha^{(q^n-1)/(q-1)}$.