Norm map is surjective for finite fields

Statement
Suppose $$K,L$$ are finite fields and $$L$$ is a (finite) field extension of $$K$$. Then, the algebraic norm map:

$$N: L^\ast \to K^\ast$$

is a surjective homomorphism.

Applications

 * Conjugacy class of elements with semisimple generalized Jordan block does not split in special linear group over a finite field