TITLE: automorphisms of a finite field
QUESTION [7 upvotes]: Let $F$ be a finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^r$ elements. Then I have proved that the map $\phi: x\mapsto x^p$ is a field automorphism of $F$. Moreover, $\phi(x)=x$ if and only if $x\in \mathbb{Z}_p$. 
I am confused on:
(1). Is every field automorphism of $F$ fixing $\mathbb{Z}_p$ can be written in this form?
I do not know how to prove:
(2). $\phi$ is an invertible linear map on the $\mathbb{Z}_p$-vector space $F$ and determine the minimal polynomial of $\phi$ over $\mathbb{Z}_p$. 
(3). Let $K$ be a  finite field of characteristic $p$ over $\mathbb{Z}_p$ with $p^t$ elements. Then $K$ is a subfield of $F$ if and only if $t\mid r$.  
(4). If $t\mid r$, then $\phi^t: x\mapsto x^{p^t}$ is a field automorphism of $F$ such that $\phi^t(x)=x$ if and only if $x\in K$. Moreover, how about the minimal polynomial of $\phi^t$ over $K$?

REPLY [3 votes]: Basically every statement on the list 1-4 is very simple but explaining it in isolation makes no sense.  It is like solving a list of calculus problems without knowing that derivatives and integrals are related to each other.
You are probably missing several of the basic facts about finite fields, which are stated  at Wikipedia, and make the answers (and maybe the proofs) to most of your questions relatively easy.
http://en.wikipedia.org/wiki/Finite_field
Let me quote some relevant parts

The finite fields are classified by size; there is exactly one finite field up to isomorphism of size $p^k$ for each prime p and positive integer k. Each finite field of size $q$ is the splitting field of the polynomial $x^q − x$, and thus the fixed field of the Frobenius endomorphism which takes x to $x^q$. Similarly, the multiplicative group of the field is a cyclic group.
The field $F_{p^n}$ contains a copy of $F_{p^m}$ if and only if m divides n.


It may also be required to know a few simple facts about fields (not only finite ones), such as an extension of $F$ being a vector space over $F$, and the very first things from Galois theory, such as the fixed field (solutions of $\psi(x)=x$) of an automorphism being a subfield.