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The R-transform as a power map and its generalisations to higher degree

Alp Bassa, Ricardo Menares

Abstract

We give iterative constructions for irreducible polynomials over F_q of degree nt^r for all nonnegative integers r, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions x^t. The R-transform introduced by Cohen is recovered as a particular case corresponding to x^2, hence we obtain a generalization of Cohen's R-transform (t=2) to arbitrary degrees t bigger that two. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of F_q we recover and generalize a recently obtained recursive construction of Panario, Reis and Wang.

The R-transform as a power map and its generalisations to higher degree

Abstract

We give iterative constructions for irreducible polynomials over F_q of degree nt^r for all nonnegative integers r, starting from irreducible polynomials of degree n. The iterative constructions correspond modulo fractional linear transformations to compositions with power functions x^t. The R-transform introduced by Cohen is recovered as a particular case corresponding to x^2, hence we obtain a generalization of Cohen's R-transform (t=2) to arbitrary degrees t bigger that two. Important properties like self-reciprocity and invariance of roots under certain automorphisms are deduced from invariance under multiplication by appropriate roots of unity. Extending to quadratic extensions of F_q we recover and generalize a recently obtained recursive construction of Panario, Reis and Wang.

Paper Structure

This paper contains 7 sections, 12 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Composition with power functions
  3. Generalities on the action of $GL_2(\mathbb{F}_q)$ on polynomials
  4. Proof of the main theorem
  5. Coordinate changes over a quadratic extension
  6. McNay's transform
  7. A transform related to Singer groups

Key Result

Theorem 1.1

Suppose $q$ is odd and let $g(x) \in \mathbb F_q[x]$ be a monic irreducible polynomial. Assume that $g(-1)\cdot g(1)$ is not a square in $\mathbb F_q$. If $q \equiv 3 \mod 4$, assume moreover that $\deg g$ is even. Consider the sequence of polynomials $(g_m(x))_{m\geq 0}$ in $\mathbb F_q[x]$ defined Then, $g_m(x)$ is an irreducible polynomial of degree $2^m\deg g$ for all $m$.

Theorems & Definitions (18)

  • Theorem 1.1: Cohen92
  • Theorem 1.2
  • Theorem 2.1: Cohen69, Theorem 1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 8 more