Show that every finite field is of prime-power order, that is, it has a prime-power number of elements.
Here are some hints which play a role:
Hint 2: Let F be a finite field of characteristic p, then every element of F is algebraic over the prime field `ZZ_p le F`.
This hint directly supports the conceptual leap from a general finite field to viewing it as an algebraic extension of its prime field.

How it connects to the proof:
Let F be a finite field of characteristic p: This is exactly what we established in Step 1 of our strategy - that every finite field has a prime characteristic p.

Every element of F is algebraic over the prime field `ZZ_p le F`:
This is the underlying reason why F can be treated as a vector space over `ZZ_p`. If elements of F were not algebraic over `ZZ_p` , F would potentially be an infinite-dimensional vector space over `ZZ_p` , which contradict F being finite. The fact that every element is algebraic means that F is a finite-dimensional vector space over `ZZ_p`.

This hint validates the idea that we can use linar algebra concepts , like dimension and basis , to describe the size of F relative to `ZZ_p` .