Existence of submultiplicative norm on a $\mathbb{C}$-vector space

Question

This is probably a very simple and maybe elementary question but, suppose we have a finite extension of $\mathbb{C}$ say $L$, is there always a sub-multiplicative norm on $L$, seen here as a finite $\mathbb{C}$-vector space ?

Answer 1

Submultiplicativity is a property of matrix norms. In particular, "$||Av|| \leq ||A||\,||v||$" and "$||AB|| \leq ||A||\,||B||$". In your case, $A,B \in M^{n \times n}(\mathbb{C})$.

If this is unclear, you may need to provide substantially more explanation about what you do and what you do not understand.

Answer 2

Let $\mathbb{C} \subseteq L$ be a finite extension of fields. Then the extension $\mathbb{C} \subseteq L$ is algebraic, since it is finite. But then we must have $L = \mathbb{C}$, since $\mathbb{C}$ is algebraically closed. The standard norm on $\mathbb{C}$ is submultiplicative.