Polynomial relations between polynomial roots
type: Item
instance of: scholarly article
title: Polynomial relations between polynomial roots
zbMATH Open document ID: 0835.12003
author: Gerd Baron
author: Mariusz Skałba
author: Michael Drmota
published in: Journal of Algebra
publication date: 1996-04-16T00:00:00Z
review text: Let \(f\) be a separable polynomial of degree \(n\) over a field \(K\) and let \(G\) be the Galois group of its splitting field. It has been shown by \textit{C. J. Smyth} [J. Number Theory 23, 243-254 (1986; Zbl 0586.12001)]\ that if \(G= S_n\) and the characteristic of \(K\) either is zero or exceeds \(n\), then there are no non-trivial additive relations between roots of \(f\). The authors first extend this result to cover the case when \(G\) is doubly transitive and the characteristic of \(K\) is either zero or prime to \(|G|\). If \(G\) is \(2k\)-transitive for some \(k\) (which in case \(k\geq 3\) means that \(G\) equals either \(S_n\) or \(A_n\) and in case \(k=2\) it may be also equal to one of the Mathieu groups) then the authors show under the same assumptions that the only polynomial relation between the roots which is linear in every variable must be a linear combination of the elementary symmetric functions with the possible exception of the case \(\text{char} (K)\leq k\). After presenting some examples of non- trivial relations between roots of a polynomial the question of additive and multiplicative relations between conjugates of algebraic numbers is studied in the language of group rings.
Mathematics Subject Classification ID: 12E05
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 827378
zbMATH Keywords: polynomial relations
zbMATH Keywords: polynomial roots
zbMATH Keywords: Galois group
zbMATH Keywords: conjugates of algebraic numbers
reviewed by: Władysław Narkiewicz
MaRDI profile type: Publication
full work available at URL: jabr.1995.1330
OpenAlex ID: W2050116005
DOI: 10.1006/JABR.1995.1330
Recommended article: Linear relations between roots of polynomials
Recommended article: Polynomials with nontrivial relations between their roots
Recommended article: Q3647359
Recommended article: About the Galois relation \(x_1=x_2+x_3\). Appendix by Joseph Oesterlé
Recommended article: Relations between polynomial roots
Recommended article: On the galois group off(xd)
Recommended article: The linear relation \(a=b+c+\dots+t\) between the roots of a polynomial
Recommended article: Q3360986
Recommended article: On the Galois group of \(x^ p + p^ tb(x+1) = 0\)
Recommended article: Specht modules and resolvents of algebraic equations
description: scientific article; zbMATH DE number 827378