On polynomials with roots modulo almost all primes

Christian Elsholtz; Benjamin Klahn; Marc Technau

doi:10.4064/aa220407-9-7

On polynomials with roots modulo almost all primes

Christian Elsholtz, Benjamin Klahn, Marc Technau

Institute of Analysis and Number Theory (5010)

Research output: Contribution to journal › Article › peer-review

Abstract

Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials h for which there is an irreducible monic quadratic g such that the product gh is exceptional. We construct exceptional polynomials with all factors of the form Xp−b with p prime and b square-free.

Original language	English
Pages (from-to)	251-263
Number of pages	13
Journal	Acta Arithmetica
Volume	205
Issue number	3
DOIs	https://doi.org/10.4064/aa220407-9-7
Publication status	Published - 2022

Keywords

Galois group
polynomial
reduction modulo p
root

ASJC Scopus subject areas

Algebra and Number Theory

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10.4064/aa220407-9-7

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title = "On polynomials with roots modulo almost all primes",

abstract = "Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials h for which there is an irreducible monic quadratic g such that the product gh is exceptional. We construct exceptional polynomials with all factors of the form Xp−b with p prime and b square-free.",

keywords = "Galois group, polynomial, reduction modulo p, root",

author = "Christian Elsholtz and Benjamin Klahn and Marc Technau",

note = "Funding Information: Both the first-and last-named author were supported by a joint FWF– ANR project {\textquoteleft}ArithRand{\textquoteright} (FWF I 4945-N and ANR-20-CE91-0006). The first-and second-named authors acknowledge support of the Austrian Science Fund (FWF), project number W1230.",

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T1 - On polynomials with roots modulo almost all primes

AU - Elsholtz, Christian

AU - Klahn, Benjamin

AU - Technau, Marc

N1 - Funding Information: Both the first-and last-named author were supported by a joint FWF– ANR project ‘ArithRand’ (FWF I 4945-N and ANR-20-CE91-0006). The first-and second-named authors acknowledge support of the Austrian Science Fund (FWF), project number W1230.

PY - 2022

Y1 - 2022

N2 - Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials h for which there is an irreducible monic quadratic g such that the product gh is exceptional. We construct exceptional polynomials with all factors of the form Xp−b with p prime and b square-free.

AB - Call a monic integer polynomial exceptional if it has a root modulo all but a finite number of primes, but does not have an integer root. We classify all irreducible monic integer polynomials h for which there is an irreducible monic quadratic g such that the product gh is exceptional. We construct exceptional polynomials with all factors of the form Xp−b with p prime and b square-free.

KW - Galois group

KW - polynomial

KW - reduction modulo p

KW - root

UR - http://www.scopus.com/inward/record.url?scp=85140798855&partnerID=8YFLogxK

U2 - 10.4064/aa220407-9-7

DO - 10.4064/aa220407-9-7

M3 - Article

AN - SCOPUS:85140798855

SN - 0065-1036

VL - 205

SP - 251

EP - 263

JO - Acta Arithmetica

JF - Acta Arithmetica

IS - 3

ER -

On polynomials with roots modulo almost all primes

Abstract

Keywords

ASJC Scopus subject areas

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