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On decomposition of sub-linearised-polynomials

Part of: Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  09 April 2009

Robert S. Coulter ,
George Havas  and
Marie Henderson
Robert S. Coulter
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716–2553, USA e-mail: coulter@math.udel.edumarie@math.udel.edu
George Havas
Affiliation:
Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland, Qld 4072, Australia e-mail: havas@itee.uq.edu.au
Marie Henderson
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716–2553, USA e-mail: coulter@math.udel.edumarie@math.udel.edu
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Abstract

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We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

MSC classification

Secondary: 11T55: Arithmetic theory of polynomial rings over finite fields 16H05: Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 317 - 328
Copyright
Copyright © Australian Mathematical Society 2004

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