Definition:Field Norm/Definition 2

Definition

Let $K$ be a field and $L / K$ a finite field extension of $K$.

Let $\alpha\in L$.

Let $L / K$ be Galois.

By Finite Field Extension has Finite Galois Group, the Galois group $\map {\operatorname{Gal}} {L / K}$ is finite.

The field norm $\map {N_{L / K}} \alpha$ of $\alpha$ is $\ds \prod_{\sigma \mathop \in \map {\operatorname{Gal}} {L / K}} \map \sigma \alpha$.

Sources

• 2010: Joseph J. Rotman: Advanced Modern Algebra (2nd ed.) Chapter $3$. Galois Theory: $\S3.2$: Fundamental Theorem of Galois Theory