TITLE: A field of order $32$
QUESTION [15 upvotes]: I was working on this problem from an old qual exam and here is the 
question. In particular this is not for homework.

True or False: There are no fields of order 32. Justify your answer.

Attempt: From general theory I know that any finite field has prime power order
and conversely given any prime power there exists a finite field of that order. So of course such fields exist. But now I need to explicitly construct such a field.
If I could somehow construct $\mathbb{Z}_2[x]/(p(x))$ where $p(x)$ is a polynomial of degree 5 which is irreducible over $Z_2$ I am done. But wait, how do I come up with a degree 5 polynomial that is irreducible over $Z_2$. My normal methods don't work here because $p(x)$ does not have order 2 or 3. In which case it is easy to check for irreducibility.
My question is in these kinds of situations, is there a general way to proceed.
Note: I have not learnt Galois' theory or anything like that. Does this problem require more machinery to solve?
Please help.

REPLY [3 votes]: Let $p$ be prime. Consider the polynomial $f(x) = x^{p^n} - x$ over $\mathbb F_p$. Its derivative is $f'(x) = -1 \ne 0$; hence $f(x)$ is separable. Therefore it has $p^n$ roots. Let $\alpha$, $\beta$ be two roots of $f(x)$. By using the Frobenius map $x \mapsto x^p$ we see that $a^{-1}$, $\alpha + \beta$ and $\alpha \beta$ are also roots of $f(x)$. Thus, the set of roots of $f(x)$ is closed under addition, multiplication and inversion. Hence, the splitting field consists entirely of the $p^n$ roots.
This shows that for any prime $p$ and any positive integer $n$, a finite field of order $p^n$ exists.
(In fact, this field is unique up to an isomorphism. This can be proved using the facts that the splitting field is unique up to an isomorphism, and the multiplicative group of a finite field is cyclic.)

Conversely, let $\mathbb F$ be a finite field. Let $\mathbb F_p$ be its prime subfield. We have $[\mathbb F : \mathbb F_p] = n$ for some positive integer $n$. Thus the order of $\mathbb F$ is $p^n$.
This shows that all finite fields have order $p^n$ for some prime $p$ and positive integer $n$.