Bhargava factorials and irreducibility of integer-valued polynomials

The ring of integer-valued polynomials over a given subset $S$ of $\mathbb{Z}$ (or $\mathrm{I}\mathrm{n}\mathrm{t}(S,\mathbb{Z}))$ is defined as the set of polynomials in $\text{Q}[x]$ which maps $S$ to $\mathbb{Z}$. In factorization theory, it is crucial to check the irreducibility of a polynomial. In this article, we make Bhargava factorials our main tool to check the irreducibility of a given polynomial $f\in \mathrm{I}\mathrm{n}\mathrm{t}(S,\mathbb{Z}))$. We also generalize our results to arbitrary subsets of a Dedekind domain.