The index of nonmonic polynomials
type: Item
instance of: scholarly article
title: The index of nonmonic polynomials
zbMATH Open document ID: 1020.11065
DOI: 10.1016/S0019-3577(01)80039-2
author: Denis Simon
published in: Indagationes Mathematicae. New Series
publication date: 2003-06-05T00:00:00Z
review text: Let \(R \subset \mathcal R\) be integral domains and \(P = a_0 x^n + \dots +a_n\) a polynomial over \(R\). Put \(P_0=1\) and \(P_i=a_0 x^i + \dots + a_i\) for \(1 \leq i \leq n-1\), assume that \(\theta \in \mathcal R\) is a root of \(P\) and define \(R[P_k(\theta)]\), the invariant ring of \(P\), as the \(R\)-module generated by \(P_i(\theta),\;0 \leq i \leq n-1\); these ideas go back to R. Dedekind and A. Hurwitz. If \(P\) is irreducible, the discriminant of \(R[P_k(\theta)]\) over \(R\) equals \(\text{disc} (P)\). With these ideas the author generalizes the notion of index to nonintegral elements of an algebraic number field \(K\) and asks whether there exists some \(\theta \in K\) with index \(1\), i.e. for which \(\theta \) does \(\mathbb Z [P_k(\theta)]\) equal the ring of integers of \(K\). For cubic number fields this question is answered affirmatively (in contrast to the result for the classical index), for higher degrees such a \(\theta\) need no longer exist. Generalizing a result of \textit{M.-N. Gras} [J. Number Theory 23, 347--353 (1986; Zbl 0564.12008)] it is shown that almost all cyclic extensions of \(\mathbb Q\) do not have elements with index \(1\). Furthermore it is shown that with increasing conductor this index goes to \(\infty\).
reviewed by: Günter Lettl
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R20
Mathematics Subject Classification ID: 13B25
zbMATH DE Number: 1893618
zbMATH Keywords: invariants
zbMATH Keywords: cyclic number field
zbMATH Keywords: discriminant
MaRDI profile type: Publication
cites work: Finiteness Theorems for Binary Forms with Given Discriminant
cites work: Q3139838
cites work: Reduction of Binary Cubic and Quartic Forms
cites work: On the density of discriminants of cubic fields. II
cites work: Computing integral points on elliptic curves
cites work: Q4510772
cites work: Q5330699
cites work: Q5734388
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description: scientific article; zbMATH DE number 1893618