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ON REGULAR GROUPS AND FIELDS

  • TOMASZ GOGACZ (a1) and KRZYSZTOF KRUPIŃSKI (a2)

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 nontrivial conjugacy class, and we notice that a classical group with one nontrivial conjugacy class is not quasi-minimal, because the centralizers of all elements are uncountable. Then, we construct a group of cardinality ω1 with only one nontrivial conjugacy class and such that the centralizers of all nontrivial elements are countable.

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