Counting polynomials over finite fields with prescribed properties — specified degree, specified splitting behavior, specified Galois group — is a classical problem in algebraic number theory. The standard approach uses the Chebotarev density theorem or its finite-field analogues, which give asymptotic counts as the field size grows.
The paper introduces a different method: counting via Galois actions on root subsets. Instead of analyzing the polynomial directly, analyze the action of the Galois group on the set of roots. Each polynomial determines a partition of its roots into orbits under the Galois action, and the partition type encodes the polynomial's splitting behavior. The counting problem reduces to enumerating Galois-compatible partitions.
The method is exact, not asymptotic. For any finite field and any degree, the count of polynomials with a given splitting type is expressed as a sum over conjugacy classes of the symmetric group, weighted by the cycle index of the Galois action. The formula is explicit enough for computation and reveals structural features that the asymptotic approach obscures.
The reframing: a polynomial over a finite field is not just a function — it's a Galois representation encoded as a partition. Counting polynomials by splitting type is counting partitions by symmetry type, and the symmetry is the Galois group's action on roots. The polynomial's algebraic properties (irreducibility, splitting pattern) become combinatorial properties (orbit structure, cycle type) under this translation. Number theory becomes combinatorics viewed through the lens of symmetry.