Fix $H$ a subgroup of $S_n$, and let $X$ be some variety parametrizing polynomials according to their coefficients.

There are two definitions of rational point, an extrinsic one--all coordinates or all ratios of coordinates are equivalent--and an intrinsic one stated in terms of the residue field of a point. We show how this second definition takes what would otherwise be a very mysterious relationship between polynomials with Galois group contained in $H$ and rational points on a cover of $X$, and turns it into a very natural correspondence. This correspondence was used in a 2002 paper by Elkies and Bruin, but the proof that will be described is in essence my own (with much credit due in the way of guidance to my undergrad advisor). Applications include restrictions on the frequency of small Galois groups in one-dimensional families of equivalence classes of polynomials, as a consequence Faltings' Theorem.

Suggested background: basic familiarity with Galois theory and algebraic geometry, will give brief overview of schemes