Fractalized cyclotomic polynomials

David P. Roberts

doi:10.1090/S0002-9939-07-08629-7

Fractalized cyclotomic polynomials

David P. Roberts

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

2 Scopus citations

Abstract

For each prime power p^m, we realize the classical cyclotomic polynomial Φ_p^m(x) as one of a collection of 3 ^m different polynomials in Z[x]. We show that the new polynomials are similar to Φ_p^m(x) in many ways, including that their discriminants all have the form±p^c. We show also that the new polynomials are more complicated than Φ_p^m(x) in other ways, including that their complex roots are generally fractal in appearance.

Original language	English (US)
Pages (from-to)	1959-1967
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	135
Issue number	7
DOIs	https://doi.org/10.1090/S0002-9939-07-08629-7
State	Published - Jul 2007

Keywords

Cyclotomic polynomial
Discriminant
Fractal
Galois

Access

10.1090/S0002-9939-07-08629-7

OpenUrl availability

Full text

Cite this

@article{939690a6752146e4a9b89e798db5e2c6,

title = "Fractalized cyclotomic polynomials",

abstract = "For each prime power pm, we realize the classical cyclotomic polynomial Φpm(x) as one of a collection of 3 m different polynomials in Z[x]. We show that the new polynomials are similar to Φpm(x) in many ways, including that their discriminants all have the form±pc. We show also that the new polynomials are more complicated than Φpm(x) in other ways, including that their complex roots are generally fractal in appearance.",

keywords = "Cyclotomic polynomial, Discriminant, Fractal, Galois",

author = "Roberts, {David P.}",

year = "2007",

month = jul,

doi = "10.1090/S0002-9939-07-08629-7",

language = "English (US)",

volume = "135",

pages = "1959--1967",

journal = "Proceedings of the American Mathematical Society",

issn = "0002-9939",

publisher = "American Mathematical Society",

number = "7",

}

TY - JOUR

T1 - Fractalized cyclotomic polynomials

AU - Roberts, David P.

PY - 2007/7

Y1 - 2007/7

N2 - For each prime power pm, we realize the classical cyclotomic polynomial Φpm(x) as one of a collection of 3 m different polynomials in Z[x]. We show that the new polynomials are similar to Φpm(x) in many ways, including that their discriminants all have the form±pc. We show also that the new polynomials are more complicated than Φpm(x) in other ways, including that their complex roots are generally fractal in appearance.

AB - For each prime power pm, we realize the classical cyclotomic polynomial Φpm(x) as one of a collection of 3 m different polynomials in Z[x]. We show that the new polynomials are similar to Φpm(x) in many ways, including that their discriminants all have the form±pc. We show also that the new polynomials are more complicated than Φpm(x) in other ways, including that their complex roots are generally fractal in appearance.

KW - Cyclotomic polynomial

KW - Discriminant

KW - Fractal

KW - Galois

UR - http://www.scopus.com/inward/record.url?scp=77950637228&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77950637228&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-07-08629-7

DO - 10.1090/S0002-9939-07-08629-7

M3 - Article

AN - SCOPUS:77950637228

SN - 0002-9939

VL - 135

SP - 1959

EP - 1967

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 7

ER -

Fractalized cyclotomic polynomials

Abstract

Keywords

Access

OpenUrl availability

Other files and links

Fingerprint

Cite this