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.

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 [x]. In this paper, we replace [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 [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.

Received:
Accepted:
Revised after acceptance:
Published online:
DOI: 10.5802/jtnb.866
Lola Thompson 1

1 Department of Mathematics Oberlin College Oberlin, OH 44074 USA
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Lola Thompson. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.866/

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