AI 中文总结
本文给出有限循环半域同伦的完全分类及自同构群的完全确定，解决了长期未决问题，还对通过斜多项式构造的最大秩距离码进行分类并描述其自同构群，推进了相关领域研究。
AI 中文摘要
1960年Hughes和Kleinfeld首次构造了有限循环半域的特殊情况，随后Sandler于1962年和Knuth于1965年也进行了构造。1966年Petit引入了循环半域的一般构造，1989年Jha和Johnson从不可约半线性变换的角度重新发现。自1962年Sandler的基础工作以来，循环半域自同构群的完全确定和该族同伦问题的完全解决一直是长期未解决的问题。本文给出了循环半域同伦的完全分类及其自同构群的完全确定，还对通过斜多项式构造的最大秩距离（MRD）码进行了分类并描述了其自同构群。
英文摘要
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.
