Dork Scratchings

Credit: Teodomiro, wikipedia.org The ancient Greeks were obsessed with ruler and compass constructions, and they had a lot of successes.  They bisected angles, constructed pentagons, and much more.  One thing that eluded them was finding a general method for trisecting an angle.  Although they could trisect certain angles, e.g. $90^{\circ}$, they tried in vain to come up with a general recipe given just three starting points. It turns out that trisecting an angle is in general impossible, but the proof that this is the case had to wait for some mathematics developed by Galois.  In this post I'll give a proof using polynomials, fields , and vector spaces . Most impossibility proofs work the same way.  First you identify some property which remains invariant with each step, and then you show that the property would need to change to get to the final state.  In this case the property is very abstract.... The Invariant Property Let $\mathbb{F}_0$ be the minimal subfield of