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 3: In essence, here's the flow and how your linear algebra basics come into play:

1. Finite field (your main object):
You start with a finite field.

2. Characteristic p and Prime Field `ZZ_p` (Step 1):
You discover that this finite field must have a prime characteristic p , and thus contains a subfield isomorphic to `ZZ_p `. This `ZZ_p` is your base field.

3. Vector Space (Step 2 & Hint 2): Because every element of F is algebraic over `ZZ_p` (Hint 2), this means F can be seen as a vector space over `ZZ_p`.
A vector space needs a set of vectors (the elements of F) and a set of scalars (the elements of `ZZ_p`) , and operations (addition and scalar multiplication) that satisfy certain axioms. Since F is a field , these properties naturally hold.

Because F is finite, it must be a finite-dimensional vector space. Let this dimension be n.