Uniqueness of finite fields with $p^n$ elements.

Question

It's well known there exists only one finite field with $p^n$ elements, up to isomorphism, where $p$ is a prime and $n \geq 1$.

Let $n = m t$, where $m, t > 1$.

Then $F = Z_p[X]/(f(X))$ is a field with $p^n$ elements, provided $f$ is some irreducible polynomial of degree $n$ in $Z_p$.

By the same token, $G = Z_p[X]/(g(X))$ is a field with $m$ elements ($g$ irreducible of degree $m$). Then $G$ has $p^m$ elements.

Finally let: $H = G[X]/(h(X))$ where $h$ is an irreducible polynomial of degree $t$ with coefficients in $G$.

Now, in my understanding $F$ and $H$ have both $p^n$ elements. So my question is:

Are $F$ and $H$ isomorphic?

Answer 1

This relies on a general fact about splitting fields.

Let $F$ be a field and $f(X)\in F[X]$ be a monic polynomial. An extension field $K$ of $F$ is a splitting field for $f$ if

1. $f(X)=(X-a_1)(X-a_2)\dots(X-a_k)$ in $K[X]$ (the roots need not be distinct);
2. $K=F(a_1,a_2,\dots,a_k)$

Theorem. If $K_1$ and $K_2$ are splitting fields of $f(X)\in F[X]$, then there exists a field isomorphism $\varphi\colon K_1\to K_2$ leaving $F$ pointwise fixed.

The proof is longish and can be found in any book on Galois theory, because it's a basic tool thereof.

Now consider the polynomial $X^{p^n}-X\in\mathbb{F}_p[X]$, where $\mathbb{F}_p$ is the $p$-element field (which is unique up to unique isomorphism).

Let $K$ be a splitting field of $f(X)$. Then $f(X)$ has $p^n$ distinct roots in $K$ (because the derivative of the polynomial is $-1$). On the other hand, the set of roots of $f(X)$ is a subfield of $K$: indeed, if $a,b$ are roots, then $$ (a+b)^{p^n}-(a+b)=a^{p^n}+b^{p^n}-a-b=0 $$ so $a+b$ is a root of $f$. Analogously $$ (ab)^{p^n}-ab=a^{p^n}b^{p^n}-ab=ab-ab=0 $$ and it's easy to check the reciprocals. Since also $0$ and $1$ are roots we're done.

Thus $K$ is the set of all roots of $f$ and therefore $|K|=p^n$.

Conversely, if $K$ is a field with $p^n$ elements, then the same argument as before shows that $X^{p^n}-X$ has $p^n$ distinct roots in $K$, so $K$ is a splitting field for $f(X)$.

Uniqueness up to isomorphism now follows from the theorem above.