Infinite families of cyclotomic function fields with any prescribed class group rank
type: Item
instance of: scholarly article
title: Infinite families of cyclotomic function fields with any prescribed class group rank
zbMATH Open document ID: 1485.11162
author: Jinjoo Yoo
author: Yoonjin Lee
published in: Journal of Pure and Applied Algebra
publication date: 2021-06-09T00:00:00Z
review text: Let \(k=\mathbb F_q(T)\) be the rational function field over the finite field \(\mathbb F_q\) of \(q\)-elements. For a polynomial \(N(t)\in \mathbb F_q[T]\), let \(\Lambda_N\) be the set of roots of \(N(\phi+\mu_T)(u)=0\) in the algebraic closure of \(k\), where \(\phi(u)=u^q,\mu_T(u)=Tu\). The field \(k(\Lambda_N)\) is called the \(N\)-th cyclotomic function field. Its maximal real subfield is denoted by \(k(\Lambda_N)^+\). In this article, for a prime divisor \(\ell\) of \(q-1\) and \(n=1,2,3\), the authors explicitly construct infinite families of the maximal real subfields of cyclotomic function fields whose ideal class groups have arbitrary \(\ell^n\)-rank, and further give a tower of cyclotomic function fields whose maximal real subfields have ideal class groups of \(\ell^n\)-ranks getting increased. The main idea is as follows. Let \(K\) be a Kummer extension over \(k\) of degree \(\ell\), \(Cl_K(\ell)\) the \(\ell\)-Sylow subgroup of the ideal class group of \(K\) and \(\sigma\) a generator of the Galois group of \(K\) over \(k\). Then \(\lambda_n=\dim_{\mathbb F_q}Cl_K(\ell)^{(\sigma-1)^{n-1}}/Cl_K(\ell)^{(\sigma-1)^n}\) is a lower bound for \(\ell^n\)-rank of \(Cl_K\). For a monic polynomial \(N\) of degree divisible by \(\ell\), the ideal class group of the Kummer extension \(k(\sqrt[\ell]N)\) is canonically isomorphic to a subgroup of the ideal class group of \(k(\Lambda_N)^+\). Therefore, the problem is reduced to construction of Kummer extensions with the desired \(\lambda_n (n=1,2,3)\). They offer algorithms to compute \(\lambda_3\) for given Kummer extensions and to give cyclotomic function fields whose \(\ell^n\)-rank of the ideal class group is greater than or equal to the given integer \(\lambda_3\). The algorithm for \(\lambda_1\) and \(\lambda_2\) has already appeared in [\textit {C.Wittmann}, Finite Fields Appl. 13, No. 2, 327--347 (2007; Zbl 1176.11061)]. They obtain the tower with the described property by using their result that for polynomi…
reviewed by: Noburo Ishii
Mathematics Subject Classification ID: 11R60
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 7357198
zbMATH Keywords: Kummer extension
zbMATH Keywords: cyclotomic function field
zbMATH Keywords: maximal real subfield
zbMATH Keywords: ideal class group
zbMATH Keywords: class group rank
MaRDI profile type: Publication
full work available at URL: j.jpaa.2020.106658
OpenAlex ID: W3135061428
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DOI: 10.1016/J.JPAA.2020.106658
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description: scientific article; zbMATH DE number 7357198