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Irreducibility of Polynomials with Square Coefficients over Finite Fields

Lior Bary-Soroker, Roy Shmueli

Abstract

We study a random polynomial of degree $n$ over the finite field $\mathbb{F}_q$, where the coefficients are independent and identically distributed and uniformly chosen from the squares in $\mathbb{F}_q$. Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches $1/n + O(q^{-1/2})$ as the field size $q$ grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.

Irreducibility of Polynomials with Square Coefficients over Finite Fields

Abstract

We study a random polynomial of degree over the finite field , where the coefficients are independent and identically distributed and uniformly chosen from the squares in . Our main result demonstrates that the likelihood of such a polynomial being irreducible approaches as the field size grows infinitely large. The analysis we employ also applies to polynomials with coefficients selected from other specific sets.

Paper Structure

This paper contains 11 sections, 13 theorems, 53 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Splitting types
  4. Discriminant
  5. Chebotarev's Density Theorem
  6. Two examples
  7. General Theorem
  8. Linearly Disjoint Extensions
  9. Proof of \ref{['thm:main']}
  10. Examples of large size sets of bounded complexity
  11. Comparison with Entin's theorem

Key Result

Theorem 1

Let $P_{{\underline{\xi}}} \in {\mathbb{F}}_q\lbrack*\rbrack{x}$ be a random polynomial as in eq:general-random-polynomial with $\xi_1, \dots, \xi_n$ uniformly distributed in $\{*\}{\alpha^2 \;\middle|\; \alpha \in {\mathbb{F}}_q} \subseteq {\mathbb{F}}_q$. Then as $q \to \infty$.

Figures (1)

  • Figure :

Theorems & Definitions (34)

  • Theorem 1
  • Theorem 2
  • Proposition 1
  • Remark 1
  • Theorem 3: Chebotarev's Density Theorem
  • proof
  • Remark 2
  • Remark 3
  • Corollary 1
  • proof
  • ...and 24 more