Subspaces and polynomial factorizations over finite fields
type: Item
instance of: scholarly article
title: Subspaces and polynomial factorizations over finite fields
zbMATH Open document ID: 0821.11061
DOI: 10.1007/BF01195333
author: Therese C. Y. Lee
author: Scott A. Vanstone
published in: Applicable Algebra in Engineering, Communication and Computing
publication date: 1995-09-27T00:00:00Z
review text: Let \(f(x)\in \mathrm{GF}(q) [x]\) be square-free of degree \(n\) over the finite field \(\mathrm{GF}(q)\) and define the Berlekamp subspace as \[ B=\{ b(x): b^q(x) \equiv b(x) \bmod f(x)\}.\] For \(q=p\) a prime, define the Niederreiter subspace as follows. Consider the differential equation \(y^{(p-1)}+ y^p=0\) where \(y^{(n)}\) denotes the \(n\)-th derivative for \(y\in K[x]\). Then for \(f(x)\in K[x]\), \(K= \mathrm{GF}(p)\) square-free, monic, of degree at least one, define the Niederreiter subspace with respect to \(f(x)\) as \[ N= \{ h(x)\in K[x]: \frac{h}{f} \text{ satisfies the equation }y^{(p-1)}+ y^p= 0\}. \] For general field order, the subspace is defined similarly using the notion of a Hasse-Teichmüller derivative. For a given polynomial and field order, the possibilities of intersection of the Berlekamp and Niederreiter subspaces is considered. The notion of a subspace that factors a polynomial \(f(x)\) completely and the notion of ``equivalent'' subspaces is introduced. The Niederreiter and Berlekamp subspaces are shown to be equivalent under a certain group action. All subspaces equivalent to the Niederreiter and Berlekamp subspaces are characterized. The orbit \(M\) under this group action which contains \(B\) and \(N\) is described.
Mathematics Subject Classification ID: 11T06
zbMATH DE Number: 750797
zbMATH Keywords: polynomial factorization
zbMATH Keywords: finite fields
zbMATH Keywords: Berlekamp subspace
zbMATH Keywords: Niederreiter subspace
zbMATH Keywords: Hasse-Teichmüller derivative
Wikidata QID: Q114234028
reviewed by: Ian Blake
MaRDI profile type: Publication
cites work: Q5550483
cites work: A new efficient factorization algorithm for polynomials over small finite fields
cites work: Factorization of polynomials and some linear-algebra problems over finite fields
full work available at URL: https://doi.org/10.1007/bf01195333
OpenAlex ID: W1993941296
description: scientific article; zbMATH DE number 750797