Nic Banks, University of Waterloo
Classification results for intersective polynomials with no integral roots
In this bald-faced attempt to practice my thesis defence, we introduce strongly intersective polynomials - polynomials with no integer roots but with a root modulo every positive integer - of degree 5-10. We start by describing their relation to Hilbert's 10th Problem and an algorithm of James Ax. These are fascinating objects which make contact with many areas of math, including permutation group theory, splitting behaviour of prime ideals in number fields, and Frobenius elements from class field theory.
In particular, we explain the computation of a list of possible Galois groups of such polynomials. We also discuss constraints on the splitting behaviour of ramified primes; in the process, we argue that intersectivity can be thought of as a property of a Galois number field, together with its set of subfields of specified degrees.
MC 5479