Galois theory without fields

My previous post described Galois connections, and how they generalize a pattern first recognized in the context of Galois theory. This pattern can extended far afield of its initial application to fields and their extensions.

For example, you could take a random variable X and think of the pair consisting of its distribution function

F: [-∞, ∞] → [0, 1]

and its quantile function

G: [0, 1] → [-∞, ∞]

as a Galois connection. It’s probably not useful to do so, but it is an example far removed from proving that there is no general formula for the roots of a quintic equation [1].

Not only can you find Galois connections in new settings, you can apply Galois theory itself in new settings. As John Baez points out here, the core idea of Galois theory is independent of its original application to field theory.

But here’s the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If you’ve got a little gadget k sitting in a big gadget K, you get a “Galois group” Gal(K/k) consisting of symmetries of the big gadget that fix everything in the little one. … Any subgroup of Gal(K/k) gives a gadget containing k and contained in K: namely, the gadget consisting of all the elements of K that are fixed by everything in this subgroup.

This came up in a conversation with a client yesterday. We’re working in a context that has very far removed from finding roots of polynomial equations. But there’s a remote possibility that we might find ideas from Galois theory useful.

The way we’re talking about Galois theory here took a century to develop. Evariste Galois certainly didn’t have this full-blown generalization in mind, though he had the germ of the idea.

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[1] There is no equation for the roots of a fifth degree polynomial that is analogous to the quadratic formula. That is, no equation that is a finite combination of roots and arithmetic operations. This was known before Galois, proved by Paolo Ruffini and Neils Henrik Abel, but Galois had an insightful way of looking at the problem.