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On the Distribution of Irreducible Trinomials

Published:2011-03-24
Printed: Dec 2011
• Igor E. Shparlinski,
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
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Abstract

We obtain new results about the number of trinomials $t^n + at + b$ with integer coefficients in a box $(a,b) \in [C, C+A] \times [D, D+B]$ that are irreducible modulo a prime $p$. As a by-product we show that for any $p$ there are irreducible polynomials of height at most $p^{1/2+o(1)}$, improving on the previous estimate of $p^{2/3+o(1)}$ obtained by the author in 1989.
 Keywords: irreducible trinomials, character sums
 MSC Classifications: 11L40 - Estimates on character sums 11T06 - Polynomials