# Irreducibility and r-th root finding over finite fields

## TL;DR

This paper explores connections between constructing r-th nonresidues and irreducible polynomials over finite fields, providing new deterministic algorithms and extending classical results like Stickelberger's Lemma.

## Contribution

It extends Stickelberger's Lemma, offers deterministic algorithms for constructing finite fields and finding r-th roots, and discusses a conjecture weaker than GRH for deterministic root finding.

## Key findings

- Deterministic poly-time construction of finite fields when m is an r-power.
- Deterministic poly-time r-th root finding when r is constant and divides gcd(m,p-1).
- Extension of Stickelberger's Lemma relating polynomial factors to nonresidues.

## Abstract

Constructing $r$-th nonresidue over a finite field is a fundamental computational problem. A related problem is to construct an irreducible polynomial of degree $r^e$ (where $r$ is a prime) over a given finite field $\mathbb{F}_q$ of characteristic $p$ (equivalently, constructing the bigger field $\mathbb{F}_{q^{r^e}}$). Both these problems have famous randomized algorithms but the derandomization is an open question. We give some new connections between these two problems and their variants. In 1897, Stickelberger proved that if a polynomial has an odd number of even degree factors, then its discriminant is a quadratic nonresidue in the field. We give an extension of Stickelberger's Lemma; we construct $r$-th nonresidues from a polynomial $f$ for which there is a $d$, such that, $r|d$ and $r\nmid\,$#(irreducible factor of $f(x)$ of degree $d$). Our theorem has the following interesting consequences: (1) we can construct $\mathbb{F}_{q^m}$ in deterministic poly(deg($f$),$m\log q$)-time if $m$ is an $r$-power and $f$ is known; (2) we can find $r$-th roots in $\mathbb{F}_{p^m}$ in deterministic poly($m\log p$)-time if $r$ is constant and $r|\gcd(m,p-1)$. We also discuss a conjecture significantly weaker than the Generalized Riemann hypothesis to get a deterministic poly-time algorithm for $r$-th root finding.