Field trace

For a finite extension L/K, Tr_{L/K}(α) is the trace of multiplication-by-α as a K-linear map.

Field trace

Definition. Let $L/K$ be a finite field extension with $n=[L:K]$ . For $\alpha\in L$ , define the $K$ -linear map

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

The trace of $\alpha$ from $L$ to $K$ is

\mathrm{Tr}_{L/K}(\alpha) \;=\; \mathrm{tr}(m_\alpha)\in K,

the ordinary trace of this linear operator.

The trace is $K$ -linear and interacts with towers via trace in towers .

See also. field norm , degree of extension .

Examples.

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