The group generated by \(x\mapsto x+1\) and \(x\mapsto x^ p\) is free
type: Item
instance of: scholarly article
title: The group generated by \(x\mapsto x+1\) and \(x\mapsto x^ p\) is free
zbMATH Open document ID: 0662.20024
DOI: 10.1016/0021-8693(88)90030-0
author: S. H. Smith
published in: Journal of Algebra
publication date: 1988-01-01T00:00:00Z
review text: Let L be an algebraically closed field of characteristic 0, and A be an additive subgroup of L. Let p be a prime number, \(\omega_{p^ n}\) denotes a primitive \(p^ nth\) root of unity in L, and let \(F={\mathbb{Q}}(A,\omega_ p,\omega_{p^ 2},...)\). Define a word to be a string \(w=w_ 1...w_ r\) made up of symbols p (``pth power''), r (``pth root''), and \(t_ a\) (``translation by a'') for \(a\in A-\{0\}\) such that no p and r are adjacent and no two symbols \(t_ a\), \(t_ b\) are adjacent. The chain associated to w means the sequence \((\zeta_ 1,...,\zeta_{n+1})\) such that \(\zeta_{i+1}=\zeta_ i+a\) if \(w_ i=t_ a\), \(\zeta_{i+1}=\zeta^ p_ i\) if \(w_ i=p\), and \(\zeta_ i=\zeta^ p_{i+1}\) if \(w_ i=r\). An algebraic extension \(K_ 2\) of a field \(K_ 1\) of degree \(| K_ 2:K_ 1| =p^ n\) is called stratified if for every m, \(0\leq m\leq n\), there exists a unique extension \(S_ n\) such that \(K_ 1\subseteq S_ m\subseteq K_ 2\) and \(| S_ m:K_ 1| =p^ m.\) The main theorem states the following. Assume \(p>2\). Assume \((\zeta_ 1,...,\zeta_{n+1})\) is the chain associated to the word \(w=w_ 1...w_ n\). If \(\zeta_ 1\) is transcendental over F, then \(F(\zeta_ 1,\zeta_{n+1})\) is a stratified algebraic extension of degree d(w) over \(F(\zeta_ 1)\) where \(d(w)=p^ b\), b is the number of r's in w. Corollary 2 states that, for any prime \(p>2\), the functions f: \(x\mapsto x+1\) and g: \(x\mapsto x^ p\) on \({\mathbb{R}}\) generate a free group under composition (this solves a problem by Harvey Friedman).
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 20E05
Mathematics Subject Classification ID: 20B27
Mathematics Subject Classification ID: 12F20
zbMATH DE Number: 4081825
zbMATH Keywords: word
zbMATH Keywords: pth power
zbMATH Keywords: pth root
zbMATH Keywords: translation
zbMATH Keywords: stratified algebraic extension
zbMATH Keywords: free group
reviewed by: Ju. I. Merzljakov
MaRDI profile type: Publication
full work available at URL: 0021-8693(88)90030-0
OpenAlex ID: W2053202682
cites work: Q3496299
cites work: On a problem of harvey friedman
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description: scientific article; zbMATH DE number 4081825