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ON THE PRODUCT OF ELEMENTS WITH PRESCRIBED TRACE

Part of: Linear incidence geometry Geometry Finite geometry and special incidence structures General field theory

Published online by Cambridge University Press:  14 May 2020

JOHN SHEEKEY Open the ORCID record for JOHN SHEEKEY [Opens in a new window] ,
GEERTRUI VAN DE VOORDE Open the ORCID record for GEERTRUI VAN DE VOORDE [Opens in a new window]  and
JOSÉ FELIPE VOLOCH Open the ORCID record for JOSÉ FELIPE VOLOCH [Opens in a new window]
JOHN SHEEKEY
Affiliation:
University College Dublin, School of Mathematics and Statistics, Science Centre, East Belfield, Dublin 4, Ireland email: john.sheekey@ucd.ie
GEERTRUI VAN DE VOORDE
Affiliation:
School of Mathematics and Statistics,University of Canterbury, Private Bag 4800,Christchurch 8140, New Zealand email: geertrui.vandevoorde@canterbury.ac.nz
JOSÉ FELIPE VOLOCH*
Affiliation:
School of Mathematics and Statistics,University of Canterbury, Private Bag 4800,Christchurch 8140, New Zealand
*
email: felipe.voloch@canterbury.ac.nz
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Abstract

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $xy$, where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.

Keywords

field extensiontracePN functionlinear set

MSC classification

Primary: 12E20: Finite fields (field-theoretic aspects)
Secondary: 51E20: Combinatorial structures in finite projective spaces 51A35: Non-Desarguesian affine and projective planes
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 112 , Issue 2 , April 2022 , pp. 264 - 288
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Copyright
© 2020 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by M. Coons

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