Describing the impossibility of trisecting the angle to high school students.

Question

Does anyone have an idea on whether it would be possible to present the proof of the impossibility of trisecting the angle (or doubling the cube, for example) in order to demonstrate the power of Galois theory, to an audience that hasn't even heard about the notion of a group before? (I am not interested in constructive approaches, I actually want to use Galois theory, directly or indirectly) Thanks, any comment is appreciated!

Answer 1

I am in fact planning on doing this! (Almost - my audience should have heard of groups, but I can't assume they remember what one is).

The proof that you can't trisect an angle in fact only really depends on the Tower Law, and the fact that the dimension of $K(\alpha)$ over $K$ is $n$ if and only if $\alpha$ satisfies an irreducible polynomial of degree $n$ with coefficients in $K$. The tower law tells you (if you set up your ruler and compass constructions axiomatically in the right way) that the only points you can construct by ruler and compass lie in field extensions of dimension $2^k$ (for some $k$) over the field you started in. Being able to trisect angles means you can construct a point with coordinate $c=\cos{\frac{\pi}{9}}$, but $c$ is a root of the irreducible degree $3$ polynomial

$$8x^3-6x-1$$ and so $\mathbb{Q}(c)$ has dimension $3$ over $\mathbb{Q}$. No groups necessary, provided you assume the two results I mentioned (and both of these results should be provable to the satisfaction of some audiences without using any group theory either).

For high school students, the trickiest thing is going to be getting them to understand the dimensions of field extensions over subfields.

The "classical" Galois theory result that really uses group theory in a crucial way is the fact that quintic polynomials cannot be generally solved in radicals.