Preservation of the residual classes numbers by polynomials
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- by Jean-Luc Chabert and Youssef Fares PDF
- Proc. Amer. Math. Soc. 139 (2011), 2423-2430 Request permission
Abstract:
Let $K$ be a global field and let $\mathcal {O}_{K,S}$ be the ring of $S$-integers of $K$ for some finite set $S$ of primes of $K$. We prove that whatever the infinite subset $E\subseteq \mathcal {O}_{K,S}$ and the polynomial $f(X)\in K[X]$, the subsets $E$ and $f(E)$ have the same number of residual classes modulo $\mathfrak {m}$ for almost all maximal ideals $\mathfrak {m}$ of $\mathcal {O}_{K,S}$ if and only if $\deg (f)=1$ when the characteristic of $K$ is 0 and $f(X)=g(X^{p^k})$ for some integer $k$ and some polynomial $g$ with $\deg (g)=1$ when the characteristic of $K$ is $p>0$.References
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Additional Information
- Jean-Luc Chabert
- Affiliation: Département de Mathématiques, LAMFA CNRS-UMR 6140, Université de Picardie, 80039 Amiens, France
- Email: jean-luc.chabert@u-picardie.fr
- Youssef Fares
- Affiliation: Département de Mathématiques, LAMFA CNRS-UMR 6140, Université de Picardie, 80039 Amiens, France
- Email: youssef.fares@u-picardie.fr
- Received by editor(s): April 11, 2010
- Received by editor(s) in revised form: July 7, 2010
- Published electronically: December 20, 2010
- Communicated by: Matthew A. Papanikolas
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2423-2430
- MSC (2010): Primary 11C08; Secondary 11A07, 11R09
- DOI: https://doi.org/10.1090/S0002-9939-2010-10696-2
- MathSciNet review: 2784807