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I am looking for a good reference for Hilbert's irreducibility theorem, and ofproperties of Hilbert sets besides Serres Lectures on The Mordell-Weil Theorem. In particular, I am interested it to the following situation:

Assuming that a variety V is defined by a polynomial H(z,t) over a field k(s), where k has finite characteristic, and s is transcendental, I'm especially interested in how many elements of form a+bs are contained in the Hilbert set of H, defined as the set of {r in k(s) such that H(z,r) is irreducible over k(s)}.

Any answers would be much appreciated!

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Try Serre, Topics in Galois Theory.

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I was just giving another reference for Hilbert irreducibility. However if you want to look into effective computations, for example for using it in conjunction with sieve methods, etc., then I do not know any reference other than Serre's Lectures on Mordell-Weil theorem. Please feel free to accept more informative answers instead, if any such turn up later. –  Anweshi May 20 '10 at 22:58

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