TITLE: Are all extensions of finite fields cyclic?
QUESTION [20 upvotes]: My book says all extensions of finite fields are cyclic, but I could not find a proof (maybe I haven't looked hard enough).  If it's straightforward, can you tell me why it's true?  Thanks :)

REPLY [26 votes]: I'm not sure that the cyclicity of the unit group of a finite field is being used here.  (Not that I have any problem using it: see e.g. Section 2 of these notes for a proof.) 
Let $K/\mathbb{F}_q$ be a field extension of degree $n$, so $\# K = q^n$.  Let $\sigma: K \rightarrow K$ be $x \mapsto x^q$. Recall:

(Lagrange's Little Theorem): Let $G$ be a finite abelian group of order $n$ and 
  $g \in G$.  Then the order of $g$ divides $n$.
The point is that LLT is a special case of Lagrange's Theorem which can be proved by the same argument which proves Fermat's Little Theorem -- i.e., the special case in which $G = \mathbb{F}_p^{\times}$.  So one need not talk about cosets and such...

Let $x \in \mathbb{F}_q$.  I claim that $x^q = x$.  This is clear if $x = 0$, and otherwise apply LLT to $x \in \mathbb{F}_q^{\times}$ to get $x^{q-1} = 1$, which implies $x^q = x$.  
Therefore $\sigma$ is an automorphism of $K/\mathbb{F}_q$.  As for its order, suppose 
$\sigma^i$ is equal to the identity: that is, for all $x \in K$, $x^{q^i} = x$.  We have $P_i(t) = t^{q^i} - t \in K[t]$ is a polynomial of degree $q^i$ over the field $K$, so by the Root-Factor Theorem (a consequence of the division algorithm for polynomials), $P_i(t)$ has at most $q^i$ roots.  It follows that the order of $\sigma$ is equal to $n = \log_q(\# K)$.  Thus the cyclic group generated by $\sigma$ is a degree $n$ subgroup of $\operatorname{Aut}(K/\mathbb{F}_q)$.  But by basic Galois theory, for an extension $K/F$ of degree $n$, we have $\# \operatorname{Aut}(K/F) \leq n$, with equality holding if and only if $K/F$ is Galois.  Therefore $K/\mathbb{F}_q$ 
is a cyclic Galois extension.  
If we like, we can now establish that there is a unique cyclic extension of degree $n$ for any $n \in \mathbb{Z}^+$: we can take 
the splitting field of $t^{q^n} - t$, and splitting fields exist and are unique up to (nonunique) isomorphism over the ground field.
It seems to me that I have not used the cyclicity of $\mathbb{F}_q^{\times}$ anywhere...