Radical Dynamical Monogenicity
Hanson Smith1
1 Department of Mathematics California State University San Marcos 333 S. Twin Oaks Valley Rd. San Marcos, CA 92096
Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 153-169

Let $a$ be an integer and $p$ a prime so that $f(x)=x^p-a$ is irreducible. Write $f^n(x)$ to indicate the $n$-fold composition of $f(x)$ with itself. We study the monogenicity of number fields defined by roots of $f^n(x)$ and give necessary and sufficient conditions for a root of $f^n(x)$ to yield a power integral basis for each $n\ge 1$. Further, we generalize these criteria to an arbitrary number field.

Soient $a$ un entier et $p$ un nombre premier tel que $f(x)=x^p-a$ est irréductible. On note $f^n(x)$ l’itéré $n$-ième de $f(x)$. Nous étudions la monogénéité des corps de nombres définis par les racines de $f^n(x)$ et donnons des conditions nécessaires et suffisantes pour qu’une racine de $f^n(x)$ génère une base entière de puissances pour chaque $n\ge 1 $. De plus, nous généralisons ces critères à un corps de nombres quelconque.

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DOI : 10.5802/jtnb.1317
Classification : 11R04, 11R21, 37P05
Keywords: Monogenic, Power integral basis, Radical extension, Iteration

Hanson Smith 1

1 Department of Mathematics California State University San Marcos 333 S. Twin Oaks Valley Rd. San Marcos, CA 92096
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Hanson Smith. Radical Dynamical Monogenicity. Journal de théorie des nombres de Bordeaux, Tome 37 (2025) no. 1, pp. 153-169. doi: 10.5802/jtnb.1317

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