[Paper Review] Special Sets of Primes in Function Fields

Julio Andrade, Steven J. Miller+2 more|arXiv (Cornell University)|Sep 22, 2013

Analytic Number Theory Research5 references3 citations

TL;DR

This paper constructs the first known nontrivial infinite F-set in function fields over finite fields, specifically for $\mathbb{F}_p[x]$ when $p \equiv 2$ or $5 \pmod{9}$. Using iterated compositions of quadratic polynomials and irreducibility criteria, it proves the existence of an infinite set of monic irreducible polynomials closed under taking factors of $P - \alpha_P$, resolving a function field analogue of a long-standing open problem in prime number theory.

ABSTRACT

When investigating the distribution of the Euler totient function, one encounters sets of primes P where if p is in P then r is in P for all r|(p-1). While it is easy to construct finite sets of such primes, the only infinite set known is the set of all primes. We translate this problem into the function field setting and construct an infinite such set in F_p[x] whenever p is equivalent to 2 or 5 modulo 9.

Motivation & Objective

• To investigate whether nontrivial infinite sets of irreducible polynomials in function fields exist, analogous to special prime sets in the integers with closure under divisors of $p-1$.
• To resolve the function field analogue of a problem in multiplicative number theory related to the Euler totient function and Lehmer's totient problem.
• To construct an explicit infinite F-set in $\mathbb{F}_p[x]$ for primes $p \equiv 2$ or $5 \pmod{9}$, where an F-set is closed under taking irreducible factors of $P - \alpha_P$ for monic irreducible $P$.
• To demonstrate that such infinite F-sets exist in function fields, providing a tractable setting to study the integer case and informing the feasibility of nontrivial infinite prime sets.

Proposed method

• Define an F-set in $\mathbb{F}_q[x]$ as a set closed under taking irreducible factors of $P - \alpha_P$ for monic irreducible $P$, where $\alpha_P$ is the constant term of $P$.
• Construct two families of polynomials: $f_\ell(x) = x^2 + x + 1$ composed with $x^{3^\ell}$, and $g_\ell(x) = x^2 - x + 1$ composed with $x^{3^\ell}$, for $\ell \geq 0$.
• Prove that $f_\ell$ and $g_\ell$ are monic and irreducible in $\mathbb{F}_p[x]$ when $p \equiv 2$ or $5 \pmod{9}$ using Theorem 2.1 from [LN], which gives conditions for irreducibility of $f(x^t)$.
• Use Lemma 2.3 to show that $f_\ell(x) - 1$ and $g\ell(x) - 1$ factor into products of lower-degree $f_k, g_k$, and linear terms, ensuring closure under the F-set condition.
• Form the set $\mathcal{F} = \{f_\ell\}_{\ell=0}^\infty \cup \{g_\ell\}_{\ell=0}^\infty \cup \{x - n\}_{n=-1}^1$, and verify it is closed under the F-set operation via factorization identities.
• Confirm $\mathcal{F}$ is nontrivial by exhibiting an irreducible polynomial not in $\mathcal{F}$, such as $x^3 + x + 1$ over $\mathbb{F}_2$ or $x+2$ over $\mathbb{F}_p$ for $p > 2$.

Experimental results

Research questions

• RQ1 Does there exist a nontrivial infinite F-set in $\mathbb{F}_p[x]$ for primes $p \equiv 2$ or $5 \pmod{9}$, where an F-set is closed under taking irreducible factors of $P - \alpha_P$?
• RQ2 Can the function field setting provide a constructive example of an infinite set of irreducible polynomials closed under the factorization condition analogous to the integer case where $p \in \mathcal{P}$ implies $r \in \mathcal{P}$ for all $r \mid (p-1)$?
• RQ3 What conditions on $p$ allow the construction of infinite F-sets using iterated compositions of quadratic polynomials?
• RQ4 Is the existence of such infinite F-sets in function fields feasible and structurally possible, and does it shed light on the integer case?
• RQ5 Can the irreducibility of $f(x^{3^\ell})$ and $g(x^{3^\ell})$ be guaranteed under specific arithmetic conditions on $p$?

Key findings

• For all primes $p \equiv 2$ or $5 \pmod{9}$, there exists a nontrivial infinite F-set in $\mathbb{F}_p[x]$.
• The constructed F-set $\mathcal{F}$ consists of iterated compositions $f_\ell(x) = (x^2 + x + 1)(x^{3^\ell})$ and $g_\ell(x) = (x^2 - x + 1)(x^{3^\ell})$, along with linear polynomials $x - n$ for $n = -1, 0, 1$.
• The polynomials $f_\ell$ and $g_\ell$ are monic and irreducible in $\mathbb{F}_p[x]$ under the given conditions on $p$, as established via Theorem 2.1 on irreducibility of $f(x^t)$.
• The factorizations of $f_\ell(x) - 1$ and $g_\ell(x) - 1$ are explicitly given and consist only of lower-degree $f_k, g_k$, and linear terms, ensuring closure under the F-set condition.
• The set $\mathcal{F}$ is proven to be infinite and nontrivial, as it excludes irreducible polynomials such as $x^3 + x + 1$ over $\mathbb{F}_2$ and $x+2$ over $\mathbb{F}_p$ for $p > 2$, which are not in the family.
• The construction relies critically on the condition $p^2 \not\equiv 1 \pmod{9}$, which excludes $p \equiv 8 \pmod{9}$, and is necessary for the applicability of Theorem 2.1.