As far as I have heard,Sophus Lie's aim was to construct an analogue of galois theory for differential galois theory. I am familiar with lie group but not with differential galois theory. What is the difference is philosophy between these 2 subjects-Lie groups and Differential Galois Theory?
Lie Groups – Differences in Philosophy Between Lie Groups and Differential Galois Theory
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I am tempted to stump for the centrality of Galois theory in modern mathematics, but I feel that this subject is too close to my own research interests (e.g., I have worked on the Inverse Galois Problem) for me to do so in a truly sober manner. So I will just make a few brief (edit: nope, guess not!) remarks:
1) Certainly when I teach graduate level classes in number theory, arithmetic geometry or algebraic geometry, I do in practice expect my students to have seen Galois theory before. I try to cultivate an attitude of "Of course you're not going to know / remember all possible background material, and I am more than willing to field background questions and point to literature [including my own notes, if possible] which contains this material." In fact, I use a lot of background knowledge of field theory -- some of it that I know full well is not taught in most standard courses, some of it that I only thought about myself rather recently -- and judging from students' questions and solutions to problems, good old finite Galois theory is a relatively known subject, compared to say infinite Galois theory (e.g. the Krull topology) and things like inseparable field extensions, linear disjointness, transcendence bases....So I think it's worth remarking that Galois theory is more central, more applicable, and (fortunately) in practice better known than a lot of topics in pure field theory which are contained in a sufficiently thick standard graduate text.
2) In response to one of Harry Gindi's comments, and to paraphrase Siegbert Tarrasch: before graduate algebra, the gods have placed undergraduate algebra. A lot of people are talking about graduate algebra as a first introduction to things that I think should be first introduced in an undergraduate course. I took a year-long sequence in undergraduate algebra at the University of Chicago that certainly included a unit on Galois theory. This was the "honors" section, but I would guess that the non-honors section included some material on Galois theory as well. Moreover -- and here's where the "but you became a Galois theorist!" objection may hold some water -- there were plenty of things that were a tougher sell and more confusing to me as a 19 year old beginning algebra student than Galois theory: I found all the talk about modules to be somewhat abstruse and (oh, the callowness of youth) even somewhat boring.
3) I think that someone in any branch of pure mathematics for whom the phrase "Galois correspondence" means nothing is really missing out on something important. The Galois correspondence between subextensions and subgroups of a Galois extension is the most classical case and should be seen first, but a topologist / geometer needs to have a feel for the Galois correspondence between subgroups of the fundamental group and covering spaces, the algebraic geometer needs the Galois correspondence between Zariski-closed subsets and radical ideals, the model theorist needs the Galois correspondence between theories and classes of models, and so forth. This is a basic, recurrent piece of mathematical structure. Not doing all the gory detail of Galois theory is a reasonable option -- I agree that many people do not need to know the proofs, which are necessarily somewhat intricate -- but skipping it entirely feels like a big loss.
(This should really be a comment I think, but I'm not highly rated enough to leave one, so please bear with me)
A Galois Theoretic condition for a polynomial in two variables to be solvable by radicals is found in the following paper: http://arxiv.org/abs/math/0305226. It seems to indicate that something similar can be done for higher variables. Perhaps I'll ask Jochen next time I see him about this.
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[This is really an extended comment on the remarks of Siegel and Khavkine, but too long for a comment box.] There is an intermediate topic which straddles the world of algebra (where differential Galois theory "really" lives, in the language of differential fields and so on) and analysis (where Lie groups live), namely the theory of linear algebraic groups. Although Lie worked from an entirely analytic viewpoint (and mainly with "group chunks", as the global perspective of Lie groups as manifolds, far from just formal group laws, only took off with the work of Weyl), it was really with the advent of the theory of matrix groups and the work of Kolchin that modern differential Galois theory (building on ideas of Liouville and Ritt) really took off. For instance, the "Lie-Kolchin" theorem about (Zariski-)connected closed subgroups of matrix groups was proved by Kolchin precisely for its applications to characterizing when a "full set of solutions" to (certain kinds of) linear ODE's could be expressed in terms of iterating the operations of forming exponentials, logarithms, and solutions to algebraic equations.
The analogy from the viewpoint of differential fields works very beautifully in the style of classical Galois theory (except that things can be even harder to compute in practice): to any linear ODE over a differential field $K$ with algebraically closed field of constants $C$ one constructs a "Picard-Vessiot" extension field $L$ of $K$ over which the ODE acquires a "full set of solutions", this being unique up to isomorphism of differential fields (analogous to splitting fields of polynomials in one variable), and the abstract automorphism group of the differential field extension $L/K$ faithfully represented on the finite-dimensional $C$-vector space $V$ of solutions (in $L$) to the given ODE turns out to be a Zariski-closed subgroup $G$ of ${\rm{GL}}(V)$ (the "differential Galois group" of the ODE). Moreover, this algebro-geometric structure on the automorphism group is intrinsic, and the "Galois correspondence" is that Zariski-closed subgroups of $G$ are in 1-1 inclusion-reversing correspondence with intermediate differential fields within $L/K$. In these terms, Kolchin related "solvability by successive exponentials, and logarithms, and algebraic equations" into "solvability of $G^0$".
The ideas introduced by Kolchin spawned techniques with $D$-modules and other aspects of "algebraic analysis" which have become important tools in geometric representation theory. Differential algebra and differential modules remains an active field, perhaps not as popular as some other areas which make use of its ideas, but it didn't take off in quite the same way as usual Galois theory simply because the kinds of things that differential Galois theory allows one to say about differential equations don't mesh as well with the kinds of things that specialists in differential equations want to know (qualitative information, non-linear phenomena, etc.). Also, relating the abstract differential field extensions to actual concrete function spaces is a tricky matter (even within the restrictive linear setting where differential Galois theory is most relevant), much like trying to relate Galois theory to the study of specific complex numbers.
Finally, and perhaps most importantly, despite the importance of compact Lie groups in particle physics, the Lie groups with the most beautiful mathematical structure (namely, semisimple ones) tend not to be the ones which are most relevant to the differential equations with symmetries that arise in a variety of physical problems. There is a beautiful book "Applications of Lie groups to differential equations" by P. Olver which does an excellent job of explaining what can be done with Lie groups in the service of symmetries in differential equations (and the Introduction explains more fully why Lie's initial dream didn't develop in quite the way he had expected).