# Contributions of Galois Theory to Mathematics

What are the major and minor contributions of Galois Theory to Mathematics? I mean direct contributions (like being aplied as it appears in Algebra) or simply by serving as a model to other theories.

I have my own answers and point of view to this question, but I think it would be nice to know your opinion too.

• An interesting question; but as it is about "points of view" and has no right answers, I've wiki-ized it. – Willie Wong Jan 29 '11 at 1:55
• Obligatory link to MO question on this subject: mathoverflow.net/questions/546/… – BBischof May 1 '11 at 16:16

The theory of covering spaces is a great example of a Galois correspondence.

• This also works in reverse: One can use the relationship between the fundamental group and covering spaces to define the fundamental group of a polynomial as its Galois group, and this leads to all sorts of things in algebraic geometry. – Dylan Wilson Jan 29 '11 at 2:14
• That's really cool. Do you have any informal reference I can check out to learn a little bit about this, or is it just sort of an underlying principle? – Aaron Mazel-Gee Jan 30 '11 at 1:25
• I know almost nothing about this, but you could start here: en.wikipedia.org/wiki/%C3%89tale_fundamental_group – Dylan Wilson Jan 30 '11 at 9:03
• Oh, someone's actually told me about this before, the etale fundamental group. But I'm confused: where do the polynomials come in -- are they just the guys defining our varieties or something? – Aaron Mazel-Gee Feb 1 '11 at 16:43

Galois theory is one of the fundamental tools in the modern theory of Diophantine equations. For example, it played a pivotal role in the proof of Mazur's theorem on the possible rational torsion points on elliptic curves over $\mathbb Q$, in Faltings's proof of Mordell's conjecture, in Wiles's proof of Fermat's Last Theorem, and in the proof by Clozel, Harris, and Taylor of the Sato--Tate conjecture.

As far as I know, the deepest contributions of Galois theory have been in this direction.

One could look at the web-page for the Galois bicentennial conference for more links and information.

The idea behind the connection between subgroups and subfields in Galois Theory has wide applications; they form an entire subject called Galois connections (same as "Galois correspondence" mentioned by Aaron Mazel-Gee). See for example the section on Galois connections in George Bergman's An Invitation to General Algebra and Universal Constructions (the files are PostScript, the index is very good, so look in the index under "Galois connection"). The structure theorem for modules over PIDs can be seen as a Galois connection, as can the Zariski correspondence between ideals of $k[x_1,\ldots,x_n]$ and varieties in $\mathbb{A}^n(k)$. (The "really interesting" parts of the Fundamental Theorem of Galois Theory, meaning the one that do not follow from general properties, are the characterizations of the point stabilizer of a subfield as a Galois Group, of the fixed sets in the field as the subfields, and of the Galois subextensions as corresponding to normal subgroups; the comparable part in the theory of the correspondence between the ideals and the varietiese is Hilbert's Nullstellensatz, which characterizes the "closed" ideals as the radical ideals).

There are several properties that one often proves from scratch in all these situations that in fact follow from the general setting. It is also mirrored to some extent when one studies group actions in general.

Galois Theory itself has lots of applications, of course: algebraic number theory uses it all the time to study rings of integers, to name one.

The Risch algorithm in Differential Galois Theory can decide if a function has an elementary primitive or not and, if it does, find it.

Galois Theory is probably the first example of a successful application of an abstract theory to solving a concrete problem.

• I disagree. Are geometry and calculus not abstract? – Dylan Wilson Jan 29 '11 at 21:06
• Maybe if it were changed from "abstract theory" to "abstract algebra" ;) – Aaron Mazel-Gee Feb 1 '11 at 7:09
• @Dylan Wilson: Perhaps it is because of the misunderstanding that we developed algebra abstractly and found some applications of it instead of the converse. – awllower Mar 6 '11 at 2:58
• I think this answer is quite correct. Galois theory is at a higher level of abstraction than calculs and geomery (at least as they were at the time Galois invented his theory), and Galois wrote specifically about the importance of replacing the (then dominant) concrete computational approach to mathematical problems by a more conceptual approach based on structural relations between different objects. – Matt E May 1 '11 at 22:24

http://www.partow.net/projects/galois/

The binary extensions of Galois fields ($\mathbb GF(2^m)$) are used extensively in digital logic and circuitry. Galois field polynomials within the branch are seen as mathematical equivalents of Linear Feed-Back Shift Register (LFSR) and operations upon elements are accomplished via bitwise operations such as xor, and, or logic. Applications within the fields of cryptography and error correcting codes use Galois fields extensively in such things as S-Box implementations (bit scramblers), strong random number generators and algebraic codes. Galois theory is used to describe and generalize results seen in these fields, for example the AES algorithm can be represented with just a few lines of mathematics using Galois theory and some other related abstract algebra.

I'm not an expert on the work of Alain Connes, but he has used Galois theory as a Leitmotiv in developing 'motivic Galois theory' to attack problems in quantum theory. Maybe someone more knowledgeable could expand on this. If you read French, you should probably take a look at his (highly informal) article 'La pensée d'Evariste Galois et le formalisme moderne', available online [http://www.alainconnes.org/docs/galoistext.pdf].

One direct and historic application is the insolvability of the quintic(Abel Ruffini theorem) . Other than that I suppose it is used in the Resolution of singularities (see the paper "resolution of singularities and modular Galois theory" by S.S.Abhyankar).The Differential version of Galois theory answers some questions about what kind of linear differential equations have solutions in the form of elementary functions.

The concept of a soluble group has its origin in Galois theory, but it is nowadays an object of study in itself and many important results in the modern theory of finite groups concern soluble groups.

If I recall correctly, Sophus Lie was so impressed by the applications of Galois theory to solvability of polynomials that he tried to construct a similar theory for ordinary differential equations; this is how the theory of Lie groups was born. (A "possible" reference for this is Borel's Essays in the history of Lie groups and algebraic groups.)