Corresponding to a given prime integer $p$, and a positive integer $n$, show that there exists a finite field consisting of $p^n = q$ roots of $x^q − x$ over $\mathbb{Z}_p$, which is determined uniquely up to an isomorphism and is denoted by $F_{p^n}$ . This field is sometimes called the Galois field $GF(p^n)$.
How to construct this field? Any reference or help is appreciated.
I think now from the comments I can give the solution.
Let $K$ be a finite field. If $charK = p$, then $K$ may be considered as a finite dimensional vector space over $\mathbb{Z}_p$. Then the dimension of $K$ over $\mathbb{Z}_p$ is finite and let $ [K : \mathbb{Z}_p] = n$.Let $B = \{x_1,x_2,...,x_n\}$ be a basis of $K$ over $\mathbb{Z}_p$ and $x \in K$. Then $x$ can be expressed as $x = a_1x_1 + a_2x_2 +···+ a_nx_n$, where $a_i \in \mathbb{Z}_p$. As $\mathbb{Z}_p$ has $p$ elements,K has at most $p^n$ elements. Since B is a basis of K, the elements $a_1x_1 + a_2x_2 +···+ a_nx_n$ are all distinct for every distinct choice of elements $a_1,a_2,...,a_n$ of $\mathbb{Z}_p$.Thus K has exactly $p^n$ elements.
Now consider the multiplicative group $K - \{0\}$ is of order $p^n − 1$. If $y$ is a non-zero element of $K$, then $y^{p^n−1} = 1$ and hence $y^{p^n}=y$.Moreover, $0^{p^n}= 0$. Thus every element of K is a root of the polynomial $x^{p^n}− x $ over $\mathbb{Z}_p$.
