Do Algebraic Extensions have Inner Products?

Question

Given that Algebraic Extensions are vector spaces over fields is it possible to define an inner product?

Answer 1

If $K \to L$ is a finite extension, there's a canonically induced bilinear form on $L$ called the trace form, defined as follows. If $a \in L$, then multiplication by $a$ defines a linear transformation $L \to L$ which has a trace $\text{tr}(a)$, and the trace form is given by

$$L \times L \ni (a, b) \mapsto \text{tr}(ab) \in K.$$

This construction shows up in algebraic number theory. However, it's not really an inner product because there isn't enough structure to state the positivity requirements on an inner product at this level of generality.