Noncommutative polynomial maps.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.16017
DOI: 10.1142/S0219498812500764
author: André Leroy
publication date: 2012-09-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.0112
Mathematics Subject Classification ID: 16S36
Mathematics Subject Classification ID: 12E05
zbMATH DE Number: 6078485
zbMATH Keywords: Ore extensions
zbMATH Keywords: finite fields
zbMATH Keywords: polynomial maps
zbMATH Keywords: factorizations of polynomials
MaRDI profile type: Publication
arXiv ID: 1208.0112
cites work: Skew-cyclic codes
cites work: Coding with skew polynomial rings
cites work: NONCOMMUTATIVE SYMMETRIC FUNCTIONS AND W-POLYNOMIALS
cites work: Pseudo-linear transformations
cites work: Vandermonde and Wronskian matrices over division rings
cites work: Wedderburn polynomials over division rings. I.
cites work: Algebraic andF-Independent Sets in 2-Firs
cites work: Theory of non-commutative polynomials
OpenAlex ID: W2963005667
published in: Journal of Algebra and its Applications
title: Noncommutative polynomial maps.
review text: Let \(p\) be a prime and \([i]=1+p+\cdots+p^{i-1}\). Suppose that \(q=p^n\) and \(\mathbb F_q\) is the finite field with \(q\) elements. Denote by \(\theta\) its Frobenius automorphism. The map \(\theta\) can be extended to the algebra of polynomials \(\mathbb F_q[t]\). Denote by \(\mathbb F_q[x^{[\;]}]\) the set of all polynomials \(\sum_{i\geqslant 0}\alpha_ix^{[i]}\). Evaluating each polynomial \(f\in\mathbb F_q[t;\theta]\) at \(x\) we obtain a polynomial \(f^{[\;]}(x)\in\mathbb F_q[x]\), namely \(f(t)(x)=f^{[\;]}(x)\). It is shown that a polynomial \(f(t)\in\mathbb F_q[t]\) is irreducible if and only if the polynomial \(f^{[\;]}(x)\in\mathbb F_q[x^{[\;]}]\subset\mathbb F_q[x]\) has no nontrivial factors from \(\mathbb F_q[x^{[\;]}]\).
reviewed by: Vyacheslav A. Artamonov
description: scientific article; zbMATH DE number 6078485