De Koninck, Jean-Marie (ed.) et al., Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4406-9/pbk). CRM Proceedings and Lecture Notes 46, 259-273 (2008).
The author proves:
Theorem 1. For every and every , there are infinitely many monic twin prime polynomial pairs in .
Theorem 2. Let be irreducible polynomials over . If is large compared to both and the sum of the degrees of the , then there is a prime dividing and an element for which every substitution with leaves all of irreducible. Explicitly, the above conclusion holds provided
Theorem 3. Fix a finite field . For each , define
and let denote the set of monic irreducibles of degree not in . Then for any
Moreover, if we assume that
where denotes the multiplicative order of modulo , then is empty for almost all (in the sense of asymptotic density).