Aesthetic proofs that involve Field Theory / Galois Theory

Question

I am preparing for an oral exam on Abstract Algebra, especially Field Theory and Galois Theory.

Now, I'm looking for some aesthetic proofs that involve Galois Theory/Field Theory for two reasons.

1. I may be asked to point out the basic concepts of Galois Theory and their fields of applications and consequences. Therefore it might be useful to know an non-standard example.

2. I am just interested in the field of Abstract Algebra and I'm looking forward to find some topics and fields, in which I can intensify my knowledge.

I know already two very basic examples:

1. The application of Field Theory to clarify the classical antique problems on Straightedge and Compass Construction (Squaring the Circle, Doubling the Cube, Angle Trisection, Construction of a regular Polygon).

2. The application of Galois Theory to determine whether a polynomial is solvable in radicals.

Which further aesthetic proofs are there, that involve the basic concepts and theorems of Field Theory / Galois Theory? Of course, the number of such proofs is huge, so I'm looking for good examples in the sense pointed out above.

Many thanks in advance.

Answer 1

I'll separate out examples in different answers as this is a big-list question. I gave what I think is the FTA proof Dietrich alludes to here. Quote:

Suppose $K$ is a Galois extension of $\mathbb{R}$. We'll aim to show that either $K = \mathbb{R}$ or $K = \mathbb{C}$. (In particular, $\mathbb{C}$ itself must therefore be algebraically closed.) Let $G$ be its Galois group and let $H$ be the Sylow $2$-subgroup of $G$.

By Galois theory, $K^H$ is an odd extension of $\mathbb{R}$. But $\mathbb{R}$ has no nontrivial odd extensions: any such extension has primitive element something with an odd degree minimal polynomial over $\mathbb{R}$, but any such polynomial has a root by the intermediate value theorem. Hence $K^H = \mathbb{R}$, or equivalently $H = G$, so $G$ has order a power of $2$.

But now $K$ is an iterated quadratic extension of $\mathbb{R}$, and it's easy to explicitly show using the quadratic formula that the only nontrivial quadratic extension of $\mathbb{R}$ is $\mathbb{C}$, which itself has no nontrivial quadratic extensions.

One of the many lovely features of this proof is that it reveals that the only analytical / topological fact you need about $\mathbb{R}$ to prove the FTA is that every polynomial of odd degree has a root. Generally speaking you can classify FTA proofs by what fundamental analytical / topological fact they use; see the old MO question listing proofs of the FTA for more. This proof also appears there (I probably learned it there!) and is attributed to Emil Artin.