# Galois Theory: necessary?

I noticed the discussion of whether the teaching of Galois Theory is necessary on MathOverflow. Here at LSE, everything we teach in mathematics should have some application to the social side of life.

Granted, there are physics applications. But are there any more social or economic applications?

Really there is a hole in what they learn in algebra here, and that is Galois Theory. But there would be an objection to teaching it if it didn't have some connection to our primary mission.

• Happily, mathematics is so plastic and versatile that you can surely find something that will fit with the mission statement. For starters, see this MSE answer: math.stackexchange.com/a/89989/43208. – Todd Trimble Jun 6 '14 at 10:25
• Maybe teach some algebraic geometry instead? Computational techniques from that area have been used in economics. For economic models based on algebraic functions, any analysis or optimization can naturally lead in that direction. – user173 Jun 6 '14 at 12:30
• Is this about undergraduate education? Graduate? Granted, there are physics applications. There are? What are they? – Ben Crowell Jun 6 '14 at 13:51
• I'm not sure that Galois Theory is the best choice if you are looking for applications to the social sciences. A quick google leads to work by Vannucci on effectivity functions (from social choice theory) that uses Galois connections, lattices. See econ-pol.unisi.it/quaderni/476.pdf as an example. You might also check out Cox's text Galois Theory and read through some of the historical motivation at the end of various sections. – Benjamin Dickman Jun 6 '14 at 21:46
• for what class? It is difficult to answer this question without a bit more context on your course content and goals. – WetlabStudent Jun 13 '14 at 5:32

Galois Theory is the place where insights from one field (structure of groups) impacts another field (study of solutions of polynomial equations). I think it's the only time undergraduate students study such a phenomenon- certainly it's a classical and profound example of the interconnectedness of ideas.

I would argue on that basis that Galois Theory is in fact be the ideal course at LSE. You see, the motto of LSE is "rerum cognoscere causas", meaning "to know the causes of things". In all other undergraduate math courses, the causes are "near", "right under our noses". But not in Galois Theory. In Galois Theory, to find the `cause' one has to look deeper! And that's the whole raison d'etre for LSE, is it not?

So I would not argue for Galois Theory on the basis of some exotic social science or economic application that might or might not exist, but rather on the basis of it being the very embodiment of the LSE motto!

• Group theory proves Fermat's little theorem in number theory. And complex analysis proves the Fundamental Theorem of Algebra. Undergraduate students often see both of these: there are interconnected ideas in the curriculum even without Galois theory. – user173 Jun 12 '14 at 10:29
• Matt F. : I would argue that neither of those are anywhere near as deep or non-obvious. Neither seems to me to represent a bridge between fields in the same sense. And both have more direct proofs. – Daniel Moskovich Jun 12 '14 at 10:33

I'm going to claim that there is no need to teach Galois theory to students at LSE. Of course there's a need for Galois Theory in the world (in the same way that literature, music and painting are necessary) but I can't see why a student in applicable mathematics would choose this over areas like: operations research, statistics, game theory, econometrics, etc.

If the question is if there are any applications of Galois theory in the "real world". I can say that Koblitz curves are used in elliptic curve cryptography (see the NIST recommendations here) and an efficient way to add a point on this curve to itself, uses an element of the Galois group (this element is known as Frobenius).

Does Galois theory really have no connection with the mission? Consider,

LSE was founded 1895 by four idealistic young Fabians for the "betterment of society". It adopted rerum cognoscere causas, which means to know the causes of things, as its motto.

You could argue, to really understand the structure of factoring polynomials you must study Galois theory. It answers the insolvability of the quintic. Moreover, applications to encryption etc.. must implicitly abound in that Galois extensions seem to be everywhere in number theory (I say this as an admitted outsider!)

For future teachers to know the causes of things they teach, Galois theory has great relevance. Surely factoring polynomials is an essential part of "real world" mathematics. In addition, I would argue that understanding the interplay between extending the number system and solving equations is quite relevant to future math teachers. Galois theory is just another chapter in that saga. The need for understanding Galois theory both in content and historical context is even more evident for the teachers of future math teachers. Is Galois theory a required math education topic? Probably not. But, if you want to be an innovator, you could study the effect of including it in your curriculum. How does it aid their future work? I'm not certain this is relevant to your students, but perhaps you could adapt the idea I sketch here. That sort of education/innovation study seems to be something you could sell at LSE.

What would a class on Field Theory look like, if designed for students concentrating in Mathematics and Economics?

(Note that at LSE, the only concentrations with math in the name are "Business Mathematics and Statistics", "Mathematics and Economics", or "Mathematics with Economics".)

As prerequisite we assume a first course in Algebra and Number Theory. From there we can see options in terms of the mathematics subject classification, e.g.:

• 12D10 (location of zeroes)
• 12F10 (Separable extensions, Galois theory)
• 12L05 (decidability)
• 12Y05 (computational aspects)
• 14P10 (semialgebraic sets)

The class could start with definitions and examples. The route to Galois theory would then focus on finite extensions of $\mathbf{Q}$, but that route is not necessary. Another route would focus on $\mathbf{R}[x]$, and real algebraic computation.

The highlights of that syllabus could include: Sturm's algorithm for finding real roots. Tarski's method for deciding the existence of points with algebraic properties. An application to the economic problem of optimizing a quadratic utility function under algebraic constraints.

A theory about the kind of knowledge that can be accessed through our human conception of an explicit solution seems to me to have applications in philosophy (and neuroscience).

Also consider differential Galois theory: although physicists derived the «equations of the world» there are fundamental obstructions to solve them «explicitly». This non-integrability property relates to what we humans call «chaos» (this property can be proved mathematically in some cases) and questions our belief that we master the world through our technology. The latter statement is clearly arguable, but it underlines the fact that despite our ability to give a precise, deterministic model of reality, we are in general no closer to predict the evolution of a system. I'm thinking of e.g. weather forecast (Lorentz's system is proved both chaotic and non-integrable), but I surmise micro-economics is somehow modelled by non-integrable equations (though I confess my total lack of knowledge in that respect).

• This is only tangentially relevant to the question. – vonbrand Jun 6 '14 at 20:48
• @vonbrand: Well, it depends on how you understand Galois theory. My viewpoint was that of a limiting theory on the solvability of equations, I wasn't referring to higher-level Galois theory in arithmetics or algebraic topology or... which are beyong the OP's intent. In that respect I claim that Galois theory of linear differential systems is easier to teach (objects are that of linear algebra) with a wider range of application. [...] – Loïc Teyssier Jun 7 '14 at 7:32
• [...] In particular it is a practical theroy together with actual tools (Maple packages etc.). I feel that educating people, who will very probably work with differential equations, and bringing them tools to actually determine if a given ODE is, say, chaotic or not can be of importance (especially considering the implications for real people of e.g. failing financial products as in 2008). – Loïc Teyssier Jun 7 '14 at 7:37
• Editing the viewpoint into the answer would markedly improve it. – Tommi Nov 19 '18 at 14:49

One possible application to consider/work toward is Algebraic Statistics. It's graduate level statistics, but maybe a focused approach can get undergraduates there in a semester or two.

• Could you expand your answer to make the connection between Galois theory and algebraic statistics more explicit? – J W Jun 14 '14 at 8:08
• ...or are you suggesting algebraic statistics as an alternative to Galois theory? – J W Jun 14 '14 at 9:29

This is an old question now, but I would toot a horn and suggest that perhaps basic finite group representation theory would fit in well, given that there are applications to noncooperative and cooperative game theory as well as social choice. There are many wonderful areas of algebra that have good connections to things LSE cares about.