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.
 A: The Risch algorithm in Differential Galois Theory can decide if a function has an elementary primitive or not and, if it does, find it.
A: Galois Theory is probably the first example of a successful application of an abstract theory to solving a concrete problem. 
A: 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.

A: 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].
A: 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.
A: 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.
A: The theory of covering spaces is a great example of a Galois correspondence.
A: 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.
A: 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.
A: 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.)
