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

Thompson, Lola

doi:10.5802/jtnb.866

Variations on a question concerning the degrees of divisors of

x^{n} - 1

Thompson, Lola ¹

¹ Department of Mathematics Oberlin College Oberlin, OH 44074 USA

Journal de théorie des nombres de Bordeaux, Volume 26 (2014) no. 1, pp. 253-267.

Abstract
Résumé

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 $ℤ [x]$ . In this paper, we replace $ℤ [x]$ with $𝔽_{p} [x]$ , where $p$ is an arbitrary-but-fixed prime. We also consider those $n$ where this condition holds for all $p$ .

Dans cet article, nous étudions une question naturelle concernant les diviseurs du polynôme $x^{n} - 1$ : à quelle fréquence $x^{n} - 1$ possède-t-il un diviseur de chaque degré entre $1$ et $n$ ? Dans un travail précédent, nous avons étudié la factorisation de $x^{n} - 1$ dans $ℤ [x]$ . Pour le présent article, nous étudions la factorisation de ce polynôme dans $𝔽_{p} [x]$ , où $p$ est un nombre premier fixé. Nous analysons aussi l’ensemble des $n$ pour lesquels $x^{n} - 1$ se factorise dans $𝔽_{p} [x]$ pour tout $p$ .

MR Zbl

DOI: 10.5802/jtnb.866

Author's affiliations:

Thompson, Lola ¹

¹ Department of Mathematics Oberlin College Oberlin, OH 44074 USA

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     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {253--267},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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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, Volume 26 (2014) no. 1, pp. 253-267. doi : 10.5802/jtnb.866. http://www.numdam.org/articles/10.5802/jtnb.866/

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