Field trace

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

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

Equivalently, if $\Omega$ is a field containing $L$ and the extension is separable (see separable extension ), then

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

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

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

2. Purely inseparable behavior (contrast). If $\mathrm{char}(K)=p$ and $L=K(t)$ with $t^p\in K$ (an inseparable extension of degree $p$), then $\mathrm{Tr}_{L/K}(t)=0$; many traces vanish in purely inseparable situations.

3. 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 .