How do you define the multiplication operation for a field extension?

Question

The addition operation can be seen to come from the fact that the extension forms a vector space over the base field. But the multiplication operation does not seem to be that obvious. Normally people just memorize the technique for multiplying two complex numbers.

Answer 1

It is common to construct the extension field as a quotient of the ring of polynomials $\mathbb{F}[x]$ modulo an ideal. So the multiplication operation is polynomial multiplication (followed by cancelling out stuff in the ideal).