Finite Fields: Root Sets, Frobenius, and Algebraic Structure

This monograph develops finite field theory through root sets, Frobenius orbits, trace duality, and cyclic modules, with applications to polynomial constructions and algebraic coding theory. Root-set descent and base extension are expressed through finite reduced algebras, while linearized polynomials connect normal bases, invariant subspaces, and linear recurrences. Coefficient pairings and derivative formulas recover information lost by trace pairings in quotient algebras with repeated factors. Cyclotomic cosets are studied through additive stabilizers, power substitution, and compatible factor lifting, including extensions to Galois rings. A central theme connects low-degree representations of subfields with conjugate root blocks, polynomial composition, and recovery from relative trace data. Return maps on root fibers determine factor degrees and organize constructions of irreducible polynomials, affine symmetries, and permutation polynomials. Applications include cyclic and constacyclic codes, BCH and Goppa distance bounds, rank-metric codes, trace codes, and Fourier support. Character orthogonality and Gauss and Jacobi sums yield counting formulas, a proof of the primitive normal basis theorem, and recurrences for extension-field character sums. Gaussian periods connect these methods with short linear relations in multiplicative subgroups. Proofs of Weil bounds for polynomial character sums combine auxiliary polynomials with generating polynomials and estimates for their reciprocal roots. The emphasis throughout is on explicit arguments connecting field structure, orbit structure, and linear dependence.