Classification Results for Intersective Polynomials With No Integral Roots

Abstract

In this thesis, we algebraically classify strongly intersective polynomials - polynomials with no integer roots but with a root modulo every positive integer - of degree 5--10. In particular, we compute a list of possible Galois groups of such polynomials. We also prove constraints on the splitting behaviour of ramified primes (i.e. primes that ramify in a splitting field of the polynomial). In the process, we show that intersectivity can be thought of as a property of a Galois number field, together with its set of subfields of specified degrees. This was achieved with characterisations of Berend-Bilu and Sonn, the latter of which we also generalise. Implementations in SageMath and GAP are provided. We also utilise Hensel's Lemma and other standard results on the local behaviour of simple field extensions.