Sunil Naik, Queen's University
On a question of Christensen, Gipson and Kulosman
The study of irreducible polynomials in various polynomial rings is an important topic in mathematics. In this context, polynomials with restricted exponents have become the focus of considerable attention in recent years. Motivated by these considerations, Matsuda introduced the ring $F[X;M]$ of polynomials with coefficients in a field $F$ and exponents in a commutative, torsion-free, cancellative (additive) monoid $M$ and began an inquiry into the irreducibility of various polynomials in these rings. For any prime $\ell$, we say that $M$ is a Matsuda monoid of type $\ell$ if for each indivisible $\alpha$ in $M$, the polynomial $X^{\alpha}-1$ is irreducible in $F[X;M]$ for any field $F$ of characteristic $\ell$.
Let $M$ be the additive submonoid of non-negative integers generated by 2 and 3. In a recent work, Christensen, Gipson, and Kulosman proved that $M$ is not a Matsuda monoid of type 2 and type 3 and they have raised the question of whether $M$ is a Matsuda monoid of type $\ell$ for any prime $\ell$. Assuming the Generalized Riemann Hypothesis (GRH), Daileda showed that $M$ is not a Matsuda monoid of any positive type. In this talk, we will discuss an unconditional proof of the above result using its connection with Artin’s primitive root conjecture.