TITLE: Aesthetic proofs that involve Field Theory / Galois Theory
QUESTION [6 upvotes]: 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.

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.
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:

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).
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.

REPLY [3 votes]: You mention the application to the classical geometrical constructions. There is a modern addition to this: origami constructions.
Cox's Galois Theory (Chapter 1o) gives the details. Here is the main result:

Theorem 10.3.6. Let $\alpha\in\mathbb{C}$ be algebraic over $\mathbb{Q}$ and let $\mathbb{Q}\subset L$ be the splitting field of the minimal polynomial of $\alpha$ over $\mathbb{Q}$. Then $\alpha$ is an origami number if and only if $[L:\mathbb{Q}]=2^a 3^b$ for some integers $a,b\geq0$

Two equivalent ways of describing the field of origami numbers: they are the numbers that can be constructed with a marked ruler; they are the numbers that can be constructed using the intersections of conics.