Splitting of Polynomial Families via Galois Theory

Abstract

We study the splitting behavior of parametrized families of polynomials over finite fields through a geometric and Galois-theoretic approach. While the underlying techniques are widely considered folklore in arithmetic geometry, they have rarely been written down explicitly. To maximize accessibility, we develop a framework based on classical Galois theory and the Chebotarev Density Theorem over an affine normal variety, avoiding the heavy machinery of Grothendieck's étale topology. The primary goal is to extend and conceptually explain a recent result by Slavov, which established the condition for square values of several polynomials over a finite field to be independent. In the case where $q\equiv 1 \pmod n$, we generalize this phenomenon to $n$-th power residues, and reframe this independence condition as the natural condition on Kummer extensions to be mutually linearly disjoint. Finally, we briefly mention how these results can be translated into the modern language of étale fundamental groups, generalizing the base to geometrically integral, normal schemes of finite type over $\mathbb{F}_q$.