Proof of Galois' theorem that there exists a field of $p^n$ elements.

Question

From Galois Theory (Rotman):

For every prime p and every positive integer n, there exists a field having exactly $p^n$ elements.

Proof. If there were a field K with $|K| = p^n = q$, then $K^* = K - \{0\}$ would be a multiplicative group of order q-1; by Lagrange's theorem $a^{q-1}=1$ for all $a \in K^*$. It follows that every element of K would be a root of the polynomial $$g(x)=x^q - x.$$

We now begin the construction. By Kronecker's theorem, there is a field E containing $\Bbb{Z}_p$ over which g(x) splits. Define $F = \{\alpha \in E: g(\alpha)=0\}$; that is, F is the set of all the roots of g(x). Since the derivative $g'(x)=qx^{q-1} - 1 = -1$ (because $q=p^n$ and E has characteristic p), Lemma 32 shows that the gcd (g,g')=1, and so g(x) has no repeated roots; that is, $|F|=q=p^n$.

We claim that F is a field, which will complete the proof...

I am a bit confused with this proof, because in order to show that F is a field with exactly $p^n$ elements, we are assuming that there exists a field K of order $p^n$. But isn't that what we are supposed to prove in the first place? In other words, how can we assume that there exists a field K of order $p^n$ for every p and n? And if we are allowed to assume that, then isn't the proof already complete?

Answer 1

I think the passage you cited would be perfectly clear if the label "Proof" came at the point where he says "We now begin the construction" and if the lines preceding that were labeled "Motivation for the proof". In other words, the business about how, if there were such a field, it would consist of roots of $g$ provides the motivation for starting the actual proof by considering a splitting field of $g$, a step that would otherwise look like magic.

Answer 2

Rotman never assumes such a field. He poses a hypothetical: what if there were such a field. What would it look like? From this hypothetical, he finds a set of criteria such a field would need to satisfy.

He then comes up with such a field by taking the roots of a polynomial. It would seem that he then shows these elements are a field (hidden by your $...$ at the end of your transcription).

This is a very common idea: focus your construction of something by finding characteristics that it must satisfy. When an engineer designs a bridge, he might say If there were a bridge here, how long would it need to be? and then proceed to design a bridge at least that long. This focuses the construction process, feels very natural, and does not assume the existence of a bridge already. And it's much better than trying random bridges (i.e. constructing random fields) hoping that one will work.