Field trace - Knowlpedia

Let $L/K$ be a finite field extension of degree $n=[L:K]$ (see degree of an extension). For $\alpha\in L$ , let

m_\alpha: L\to L,\qquad x\mapsto \alpha x,

viewed as a $K$ -linear endomorphism of the $K$ -vector space $L$ . The (field) trace of $\alpha$ from $L$ to $K$ is

\mathrm{Tr}_{L/K}(\alpha) := \mathrm{trace}(m_\alpha)\in K.

Equivalent characterizations

Equivalently, if $L/K$ is separable and $\Omega$ is an algebraically closed extension of $K$ , then

\mathrm{Tr}_{L/K}(\alpha)=\sum_{\sigma} \sigma(\alpha),

where the sum runs over all $K$ -embeddings $\sigma:L\hookrightarrow \Omega$ (counted without repetition). In particular, $\mathrm{Tr}_{L/K}$ is $K$ -linear and satisfies the tower property in trace/norm in towers for a tower of fields $K\subseteq M\subseteq L$ .

Examples

Quadratic extension. Let $L=K(\sqrt{d})$ with $\mathrm{char}(K)\neq 2$ . For $\alpha=a+b\sqrt{d}$ ,

\mathrm{Tr}_{L/K}(\alpha)=(a+b\sqrt{d})+(a-b\sqrt{d})=2a.

Purely inseparable behavior (contrast). If $\mathrm{char}(K)=p$ , $L=K(t)$ , and $t^p\in K$ with $[L:K]=p$ , then $L/K$ is purely inseparable and $\mathrm{Tr}_{L/K}(t)=0$ .

Finite fields. For $L=\mathbb{F}_{q^n}$ over $K=\mathbb{F}_q$ (a finite field extension), one has

\mathrm{Tr}_{L/K}(\alpha)=\alpha+\alpha^{q}+\alpha^{q^2}+\cdots+\alpha^{q^{n-1}},

where $x\mapsto x^q$ is a power of the Frobenius.