Galois theory of deformations of polynomials

Paul Andrew Loomis, Purdue University

Abstract

Using an explicit symplectic form, we deform a polynomial found by Abhyankar to expand its Galois group over a characteristic p field from the symplectic (isometry) group Sp(2m, q) to the symplectic similitude group GSp(2m, q). As a byproduct, we relax the hypothesis in Abhyankar's original paper that the base field k be algebraically closed, requiring only that [special characters omitted]. We also develop a more general homogenization method to deform polynomials, expanding their Galois groups over characteristic p fields from various classical isometry groups toward their corresponding similitude groups. In addition we outline a generalized iteration method with the goal of constructing a polynomial with Galois group GL(m, qn) over the field [special characters omitted]. We obtain partial results in this direction, and show explicitly that the method succeeds for m = n = q = 2.

Degree

Ph.D.

Advisors

Abhyankar, Purdue University.

Subject Area

Mathematics

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