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:

4. Basis and Elements (Step 3 & Hint 1) :
If F is an n-dimensional vectorspace over `ZZ_p` , it means there exists a basis of n elements , say
{`alpha_1 , ... , alpha_n`}.
Every vector (element of F) can be written uniquely as a linear combination of the basis vectors :
`c_1 alpha_1 + ... + c_n alpha_n` , where `c_i` are scalars fom `ZZ_p`.
Since `ZZ_p` has p elements , there are p choices for each `c_i` . Because there are n such choices (one for each element) , the total number of unique linear combinations (and thus unique elements in F) is
`p * p * ... * p` (n times) , which is `p^n` . This directly reflects the idea in Hint 1 (`q^n` where q = p).

So , the hints guide you to understand why a finite field behaves like a finite-dimensional vector space over its prime field , and why this leads to a prime-power order.