Talk:Hilbert's irreducibility theorem

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Sorry - how exactly is this supposed to be cleaned up? Charles Matthews 21:32, 27 May 2005 (UTC)

Finite fields

Does a similar theorem hold when the coefficients are not rational, but elements of some finite field?--129.70.14.128 (talk) 23:28, 19 November 2007 (UTC)

Yes, there is a similar theorem for `large' finite fields, that is a special case of Chebotarev's theorem for finite fields. Here is a precise formulation:

Let f(X,Y) be an irreducible polynomial with coefficients in a field F with q elements that is separable in Y. Denote the Galois group of f over F(X) by G. Embed G in the symmetric group via the action on the roots of f. Let p be the probability to choose an n-cycles in G. (An element of G is said to be an n-cycle if it acts transitively on the roots of f.) Then there are

$pq+O({\sqrt {q}})$

x in F such that f(x,Y) is irreducible. Here the asserted constant depends only on the Y-degree of f. --Barylior (talk) 18:26, 20 September 2009 (UTC)

Remark

The first remark ("It follows from the theorem that there are infinitely many r-tuples.") makes little sense to me. Of course, there are many r-tuples. If we only want r-tuples with the property in the theorem, then I don't see why there are "many". Maybe it is meant "it follows from the proof"?. Benji104 (talk) 07:23, 26 July 2023 (UTC)