素数维半域与Kaplansky猜想的反例

中文摘要 AI 辅助

1975年，Kaplansky猜想：在足够大的有限域上，每个五维可除代数要么是域，要么是扭域。我们否证了这一猜想。对于每个素数幂$q=p^e\equiv1\pmod3$以及每个满足$\gcd(n,6)=1$的$n\ge5$，我们构造了阶为$q^n$的半域。对于固定的$q,n$，若$p\equiv1\pmod3$，该族表示$\varphi(n)$个同位类；若$p\equiv2\pmod3$，则表示$\varphi(n)/2$个同位类，其中$\varphi$为欧拉 totient 函数。利用新的同位不变量以及我们构造的结构性质，我们证明了这些半域中没有一个与有限域或Albert广义扭域同位。特别地，五维特例给出了无穷多个两两不同位的反例，这些反例在任意大的有限域上否证了Kaplansky猜想。更一般地，对于每个素数维$n\ge5$，这些例子出现在除特征3外的任意大特征域上，这与Menichetti在1996年所断言的分类相矛盾，其证明存在漏洞。

英文摘要

In 1975, Kaplansky conjectured that every five-dimensional division algebra over a sufficiently large finite field is a field or a twisted field. We disprove this conjecture. For every prime power $q=p^e\equiv1\pmod3$ and every $n\ge5$ with $\gcd(n,6)=1$, we construct semifields of order $q^n$. For fixed $q,n$, the family represents $φ(n)$ isotopy classes if $p\equiv1\pmod3$, and $φ(n)/2$ if $p\equiv2\pmod3$, where $φ$ is Euler's totient function. Using new isotopy invariants and the structural properties of our construction, we prove that none of these semifields is isotopic to a finite field or an Albert's generalized twisted field. In particular, the five-dimensional specialization gives infinitely many pairwise nonisotopic counterexamples to Kaplansky's conjecture over arbitrarily large finite fields. More generally, for each prime dimension $n\ge5$, the examples occur over arbitrarily large fields in every characteristic other than three, contradicting the classification asserted by Menichetti in 1996, whose proof contains gaps.

发表机构

• Bolyai Institute, University of Szeged（塞格德大学博莱伊研究所）
• College of Science, National University of Defense Technology（国防科技大学理学院）

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