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I have always been interested in the Inverse Galois Problem since I read the story of Évariste Galois. I have recently finished up courses in rings, fields, groups, and finally Galois theory. I really fell in love with Galois theory in general and being able to really understand, mathematically, the statement of the Inverse Galois Problem only made me want to explore it more.

My question is: where do I go from here? What subjects are required to understand current research on the problem?

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Great! The inverse Galois problem has always fascinated me.

A list of references is given in this answer. I particularly recommend Völklein's book, which is meant as an introduction. As he mentions, the basic techniques require only knowledge of introductory Galois theory, (finite) group theory, some algebra, and complex analysis since the theory begins by having groups realized via covering spaces of Riemann surfaces. The basic approach, dating back to Hilbert, who was the first to work seriously on the problem, is the irreducibility theorem, that ensures that groups realized over $\mathbb Q(x)$ are realized over $\mathbb Q$.

Deeper results require more background: An understanding of the theory of simple groups, for example, and techniques from algebraic geometry. There is the hope that the Classification of finite simple groups will allow an "inductive" solution of the inverse Galois problem, and this has greatly influenced modern research on the question. Some results, such as Shafarevich's realization of all solvable groups over $\mathbb Q$, are more ad hoc. Shafarevich's result is essentially number theoretic, for example.

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Your link appears to be missing. –  Potato Sep 4 '13 at 2:21
I assume you mean this textbook Groups as Galois Groups by Völklein. –  Islands Sep 4 '13 at 2:23
@Islands Yes, one would think I would have mastered copy-paste by now. I think it is now fixed. –  Andrés Caicedo Sep 4 '13 at 2:44
One thing is that, due to the nature of the approach, many groups $G$ are realized without solving the basic question of which irreducible polynomials give rise to Galois extensions with $G$ as Galois group. This question is addressed in the Jensen-Ledet-Yui book, also listed in the link above. –  Andrés Caicedo Sep 4 '13 at 2:46
One cool application of geometry that you hint at here is the proof that any group is realizable over any extension of $\mathbb{C}$ of transcendence degree 1. In particular, any group is realizable over $\mathbb{C}(z)$. The question is obviously much harder for $\mathbb{Q}(z)$! –  rfauffar Sep 4 '13 at 2:49

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