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Making Pretty Pictures for Galois Theory

19 May 2021 - Tags: sage , pretty-pictures

So in my algebra class we’re doing galois theory, a subject which never seems to really click with me. I know a lot of the theorems, and I can even solve a lot of the problems, but I always feel uneasy about it. The computational problems often feel like guesswork, and the theoretical problems feel either trivial or impossible with little in between. I used to feel this way about analysis, but it stings more to be struggling so much with a subject so near to my heart.

All this to say I’ve been spending a lot of time reading about galois theory and thinking about galois theory. I found a paper1 which includes some pretty pictures of roots of polynomials and how they permute. I know that the “symmetries” we study in galois theory are more abstract than symmetries we can obviously see by plotting roots in the complex plane, but I thought it could be fun to make some pretty pictures of my own anyways. My undergraduate advisor (Klaus Sutner) loves making pictures of everything he studies, and a lot of that has rubbed off on me. You never know when some picture will exhibit some pattern which you can make mathematically precise.

Of course, once you’re making one or two sets of pictures, you might as well make a framework for making them. I also like showing off sage on this blog, so it only makes sense to put my code here for future students to play around with. Some of the arrows are a little messed up, but my basic algorithm actually makes a lot of them look quite nice ^_^.

As one quick word of caution: keep in mind that most polynomials have the whole symmetric group as their group of symmetries! So when you plot those pictures you’re likely to get factorially many in the degree! Be careful plotting any polynomials of degree $> 4$ if you want to keep the computation somewhat quick.

As a (fun?) exercise, can you modify the above code to pick a generating set for the galois group and plot the generators in red rather than blue? Or what about only plotting the generators? That might let you visualize galois groups of larger polynomials.

  1. Curtis Bright’s Computing the Galois Group of a Polynomial