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I was trying to read a paper on Inverse Galois problem . I understands what the inverse Galois problem is. It asks if every finite group is the Galois group of some extension of the rationals.

The authors says that If we can show that every extension of $\mathbb{Q}$ the specialization of a Branched cover of $P^{1}$ we can use Hilbert's irreducibility theorem to get the result.

Actually I have the folowing questions

  1. What is the progress made in showing that given a group $G$ , it is a Galois Group of some branched cover of $P^{1}$ defined over rationals.

2.What are the techniques developed to construct these coverings ?

If someone can tell me the sources from where I can read about these things in detail I will try to read on my own. I mean is there any book that talks about construting coverings of $P^{1}$ over rationals.

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You could try [Malle-Matzat], "Inverse Galois Theory".

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