Rational transformations over finite fields that are never irreducible

IF 1.2 2区数学 Q3 COMPUTER SCIENCE, THEORY & METHODS Designs, Codes and Cryptography Pub Date : 2025-04-24 DOI:10.1007/s10623-025-01591-2

Max Schulz

引用次数: 0

Abstract

Rational transformations play an important role in the construction of irreducible polynomials over finite fields. Usually, the methods involve fixing a rational function Q and deriving conditions on polynomials \(F\in \mathbb {F}_q[x]\) such that the rational transformation of F with Q is irreducible. Here we want to change the perspective and study rational functions with which the rational transformation never yields irreducible polynomials. We show that if the rational function is contained in certain subfields of \(\mathbb {F}_q(x)\) then the rational transformation with it is always reducible. This extends the list of known examples.

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有限域上从不不可约的有理变换

有理变换在有限域上不可约多项式的构造中扮演着重要角色。通常，这些方法涉及固定一个有理函数 Q，并推导出多项式 \(F\in \mathbb {F}_q[x]\) 的条件，使得 F 与 Q 的有理变换是不可还原的。这里我们想换个角度，研究有理函数的有理变换永远不会产生不可约多项式。我们证明，如果有理函数包含在 \mathbb {F}_q(x)\ 的某些子域中，那么它的有理变换总是可还原的。这扩展了已知例子的范围。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Designs, Codes and Cryptography 工程技术-计算机：理论方法

CiteScore

2.80

自引率

12.50%

发文量

157

审稿时长

16.5 months

期刊介绍： Designs, Codes and Cryptography is an archival peer-reviewed technical journal publishing original research papers in the designated areas. There is a great deal of activity in design theory, coding theory and cryptography, including a substantial amount of research which brings together more than one of the subjects. While many journals exist for each of the individual areas, few encourage the interaction of the disciplines. The journal was founded to meet the needs of mathematicians, engineers and computer scientists working in these areas, whose interests extend beyond the bounds of any one of the individual disciplines. The journal provides a forum for high quality research in its three areas, with papers touching more than one of the areas especially welcome. The journal also considers high quality submissions in the closely related areas of finite fields and finite geometries, which provide important tools for both the construction and the actual application of designs, codes and cryptographic systems. In particular, it includes (mostly theoretical) papers on computational aspects of finite fields. It also considers topics in sequence design, which frequently admit equivalent formulations in the journal’s main areas. Designs, Codes and Cryptography is mathematically oriented, emphasizing the algebraic and geometric aspects of the areas it covers. The journal considers high quality papers of both a theoretical and a practical nature, provided they contain a substantial amount of mathematics.

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