Uniqueness of finite field

Question

Assume $L$ is the algebraic closure of $\mathbb{F}_p$. Show there exists a unique finite field of cardinality $p^n$ containing $\mathbb{F}_p$. The existence is easy just have to define the splitting field of $X^{p^n}-X$. But what about uniqueness?

Answer 1

HINT: Prove that every element of such extension is a root of $x^{p^n} - x$.

Answer 2

The multiplicative subgroup of nonzero elements of a field with $p^n$ elements is an abelian group of $p^n - 1$ elements; every element is a root of $X^{p^n - 1} - 1$.