M Theory Lesson 267

Usually when playing with a category of vector spaces over a field $K$, one either puts $K = \mathbb{C}$ or one doesn't worry about the field at all. But there is a situation when finite fields are the only appropriate choice.

Recall that the adjunction from Set to Vect contains a functor $G:$ Vect $\rightarrow$ Set which sends a vector space to its underlying set. Over $\mathbb{C}$ this is clearly an infinite set. So if we wanted to restrict to FinSet, the category of finite sets, there would be no way to maintain the adjunction. On the other hand, with a finite field, although a finite dimensional vector space may be large, it is still finite.

Mathematicians sometimes say that finite fields are a lot like the complex numbers anyway. Without zero, the multiplicative structure is just like the roots of unity in the complex plane. And MUB matrices for finite dimensional Hilbert spaces in Mersenne prime dimensions only require finite fields.