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On the Number of Times a Root off(n, x)=0 Generates a Field Containing a Given Number Field

✍ Scribed by Umberto Zannier

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
234 KB
Volume
72
Category
Article
ISSN
0022-314X

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✦ Synopsis

Let f (x, y) be a polynomial with rational coefficients, and let E be a number field. We prove estimates for the number of positive integers n T such that some root : of f (n, y)=0 satisfies E/Q(:). The estimates are uniform with respect to E and, provided E satisfies natural necessary conditions, take the form c log TÂlog log T T 1Â2 , where c depends only on f. Under fairly general assumptions on f, we show that the factor T 1Â2 may be removed. We also observe how these results lead to a version of Hilbert's Irreducibility Theorem uniform over number fields.

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