Contributions of Galois Theory to Mathematics

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

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.