Irreducibility of random polynomials of large degree

We consider random polynomials with independent identically distributed coefficients with a fixed law. Assuming the Riemann hypothesis for Dedekind zeta functions, we prove that such polynomials are irreducible and their Galois groups contain the alternating group with high probability as the degree goes to infinity. This settles a conjecture of Odlyzko and Poonen conditionally on RH for Dedekind zeta functions.

Keywords

random polynomials, irreducibility, Riemann hypothesis, Dedekind zeta function, Markov chains

2010 Mathematics Subject Classification

Primary 11C08. Secondary 11M41, 60J10.

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E. B. acknowledges support from ERC Grant no. 617129 ‘GeTeMo’. P. V. acknowledges support from the Royal Society.

Received 31 October 2018

Accepted 22 September 2019

Published 19 December 2019