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AN EFFECTIVE ANALYTIC FORMULA FOR THE NUMBER OF DISTINCT IRREDUCIBLE FACTORS OF A POLYNOMIAL

Part of: Elementary number theory General field theory Polynomials and matrices

Published online by Cambridge University Press:  09 December 2021

STEPHAN RAMON GARCIA Open the ORCID record for STEPHAN RAMON GARCIA [Opens in a new window] ,
ETHAN SIMPSON LEE ,
JOSH SUH Open the ORCID record for JOSH SUH [Opens in a new window]  and
JIAHUI YU
STEPHAN RAMON GARCIA*
Affiliation:
Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA
ETHAN SIMPSON LEE
Affiliation:
School of Science, UNSW Canberra at the Australian Defence Force Academy, Northcott Drive, Campbell ACT 2612, Australia e-mail: [email protected]
JOSH SUH
Affiliation:
Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA
JIAHUI YU
Affiliation:
Department of Mathematics, Pomona College, 610 N. College Ave., Claremont, CA 91711, USA
*
e-mail: [email protected]
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Abstract

We obtain an effective analytic formula, with explicit constants, for the number of distinct irreducible factors of a polynomial $f \in \mathbb {Z}[x]$ . We use an explicit version of Mertens’ theorem for number fields to estimate a related sum over rational primes. For a given $f \in \mathbb {Z}[x]$ , our result yields a finite list of primes that certifies the number of distinct irreducible factors of f.

Keywords

polynomialirreduciblenumber fieldMertens’ theoremprimeprime idealdiscriminant

MSC classification

Primary: 11C08: Polynomials
Secondary: 12E05: Polynomials (irreducibility, etc.) 11A41: Primes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 113 , Issue 3 , December 2022 , pp. 339 - 356
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Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by Michael Coons

SRG was supported by NSF grants DMS-2054002 and DMS-1800123.

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