Order of finite fields is $p^n$

Question

Let $F$ be a finite field. How do I prove that the order of $F$ is always of order $p^n$ where $p$ is prime?

Answer 1

1. Prove that the smallest multiple $m$ of 1 that gives zero has to be a prime. (Otherwise there are divisors of $m$ which are then divisors of zero.)

2. Prove that a field is a vector space over a subfield.

3. Count the elements of the field if the dimension of this vector space is $n$.