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arXiv · 2607.19172

On the autotopism groups and the equivalence of finite cyclic semifields

Paolo Santonastaso·Yue Zhou
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Abstract

Special cases of finite cyclic semifields were first constructed by Hughes and Kleinfeld in 1960, and later by Sandler in 1962 and Knuth in 1965. The general construction of cyclic semifields was subsequently introduced by Petit in 1966, and later rediscovered from the perspective of irreducible semilinear transformations by Jha and Johnson in 1989. Since Sandler's foundational work in 1962, the complete determination of the autotopism groups of cyclic semifields and the full resolution of the isotopy problem for this family have remained long-standing open problems. The most significant advances in determining these autotopism groups are due to Dempwolff in 2011, who left open the case in which the field extension degree strictly divides the degree of the polynomial defining the semifield. In this paper, we provide a complete classification of cyclic semifields up to isotopy, together with the full determination of their autotopism groups, thereby closing the remaining cases left open by Dempwolff. Since cyclic semifields arise as a special instance of a broader family of maximum rank distance (MRD) codes constructed via skew polynomials, our methods also yield a complete classification of these MRD codes up to linear and semilinear equivalence over the prime field, together with an explicit description of their full automorphism groups.

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Paolo Santonastaso, Yue Zhou. 2026-07-21. On the autotopism groups and the equivalence of finite cyclic semifields. https://arxiv.org/abs/2607.19172

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