Aesthetic proofs that involve Field Theory / Galois Theory

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.

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