A closer look at some cyclic semifields

Abstract

We prove that the choice of the automorphism $\sigma$ in the construction of a Petit semifield is important when counting isomorphism classes. Different choices of generators $\sigma$ of the Galois group of a cyclic field extension $K/F$ of degree $n$ can produce non-isomorphic proper nonassociative Petit algebras $K[t;\sigma ]/K[t;\sigma ](t^m-a)$ when $n\nmid m$, or when $n=m$ and $a$ does not lie in the fixed field of $\sigma$. In particular, this is the case when $F$ is a finite field. Thus there are $\phi (n)$ families of these semifields (where $\phi$ is the Euler function); one family for each distinct generator $\sigma$ involved in their construction, and no algebra in one family is isomorphic to one in another family. We then turn to the isotopy classes of these algebras. When the order of $\sigma$ is greater or equal to the degree $m$, and whenever $m\ge 3$ when $n>m$, two proper nonassociative Petit division algebras $K[t;\sigma ]/K[t;\sigma ](t^m-a_1)$ and $K[t;\sigma ]/K[t;\sigma ](t^m-a_2)$ which are isotopic are indeed already isomorphic, whenever $\mathrm{Aut}(K)$ is abelian. Under the same assumptions on $n$ and $m$, two such semifields constructed from the same generator $\sigma$ of $\mathrm{Gal}({\mathbb{F}}_{q^n}/{\mathbb{F}}_q)$ are thus isomorphic whenever they are isotopic. For $n=m$, we show that each isotopy class of Petit semifields $K[t;\sigma ]/K[t;\sigma ]f$ contains a proper nonassociative cyclic algebra $(K/F,\sigma ,a)$. We then count the isotopy classes of all Petit semifields when $n>2$. We also present some upper bounds, counts, and parametrizations for different equivalence and isomorphism classes. Most of our results are proved in all generality for any cyclic Galois field extension.

New version has a new section where the number of isotopy classes for Petit semifields is counted