# Variations on a question concerning the degrees of divisors of ${x}^{n}-1$

• [1] Department of Mathematics Oberlin College Oberlin, OH 44074 USA
• Volume: 26, Issue: 1, page 253-267
• ISSN: 1246-7405

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## Abstract

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In this paper, we examine a natural question concerning the divisors of the polynomial ${x}^{n}-1$: “How often does ${x}^{n}-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when ${x}^{n}-1$ is factored in $ℤ\left[x\right]$. In this paper, we replace $ℤ\left[x\right]$ with ${𝔽}_{p}\left[x\right]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.

## How to cite

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Thompson, Lola. "Variations on a question concerning the degrees of divisors of $x^n-1$." Journal de Théorie des Nombres de Bordeaux 26.1 (2014): 253-267. <http://eudml.org/doc/275785>.

@article{Thompson2014,
abstract = {In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb\{Z\}[x]$. In this paper, we replace $\mathbb\{Z\}[x]$ with $\{\mathbb\{F\}\}_p[x]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.},
affiliation = {Department of Mathematics Oberlin College Oberlin, OH 44074 USA},
author = {Thompson, Lola},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
month = {4},
number = {1},
pages = {253-267},
publisher = {Société Arithmétique de Bordeaux},
title = {Variations on a question concerning the degrees of divisors of $x^n-1$},
url = {http://eudml.org/doc/275785},
volume = {26},
year = {2014},
}

TY - JOUR
AU - Thompson, Lola
TI - Variations on a question concerning the degrees of divisors of $x^n-1$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2014/4//
PB - Société Arithmétique de Bordeaux
VL - 26
IS - 1
SP - 253
EP - 267
AB - In this paper, we examine a natural question concerning the divisors of the polynomial $x^n-1$: “How often does $x^n-1$ have a divisor of every degree between $1$ and $n$?” In a previous paper, we considered the situation when $x^n-1$ is factored in $\mathbb{Z}[x]$. In this paper, we replace $\mathbb{Z}[x]$ with ${\mathbb{F}}_p[x]$, where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$.
LA - eng
UR - http://eudml.org/doc/275785
ER -

## References

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1. D. Dummit, R. Foote, Abstract algebra. John Wiley & Sons, Inc., USA, 2004. Zbl1037.00003MR2286236
2. P. Pollack, Not always buried deep: a second course in elementary number theory. Amer. Math. Soc., Providence, 2009. Zbl1187.11001MR2555430
3. E. Saias, Entiers à diviseurs denses. I., J. Number Theory 62 (1997), 163–191. Zbl0872.11039MR1430008
4. B. M. Stewart, Sums of distinct divisors, Amer. J. Math. 76 no. 4 (1954), 779–785. Zbl0056.27004MR64800
5. G. Tenenbaum, Lois de répartition des diviseurs, 5, J. London Math. Soc. (2) 20 (1979), 165–176. Zbl0422.10050MR551441
6. G. Tenenbaum, Sur un problème de crible et ses applications, Ann. Sci. École Norm. Sup. (4) 19 (1986), 1–30. Zbl0599.10037MR860809
7. L. Thompson, Polynomials with divisors of every degree, J. Number Theory 132 (2012), 1038–1053. Zbl1287.11113MR2890525
8. L. ThompsonOn the divisors of ${x}^{n}-1$ in ${F}_{p}\left[x\right]$, Int. J. Number Theory 9 (2013), 421–430. Zbl1271.11094MR3005557
9. L. ThompsonProducts of distinct cyclotomic polynomials, Ph.D. thesis, Dartmouth College, 2012.

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