Field norm

Summary

Let $L/K$ be a finite extension and let $\alpha \in L$. Multiplication by $\alpha$ is a $K$-linear map of $L$, and the field norm $N_{L/K}(\alpha)$ is its determinant. It is a multiplicative map $L \to K$, and it is nonzero exactly on the nonzero elements. For a separable extension it is the product of the images of $\alpha$ under all embeddings of $L$ into an algebraic closure. Norms measure the size of algebraic numbers, detect units in rings of integers, and appear in Hilbert's Theorem 90.