On regular groups and fields

Abstract

Regular groups and fields are common generalizations of minimal and quasi-minimal groups and fields, so the conjectures that minimal or quasi-minimal fields are algebraically closed have their common generalization to the conjecture that each regular field is algebraically closed. Standard arguments show that a generically stable regular field is algebraically closed. Let $K$ be a regular field which is not generically stable and let $p$ be its global generic type. We observe that if $K$ has a finite extension $L$ of degree $n$, then $p^{(n)}$ has unbounded orbit under the action of the multiplicative group of $L$. Known to be true in the minimal context, it remains wide open whether regular, or even quasi-minimal, groups are abelian. We show that if it is not the case, then there is a counter-example with a unique non-trivial conjugacy class, and we notice that a classical group with one non-trivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then we construct a group of cardinality $ω_1$ with only one non-trivial conjugacy class and such that the centralizers of all non-trivial elements are countable.