Around Podewski's conjecture

TL;DR

This paper investigates Podewski's conjecture that minimal fields are algebraically closed, reducing it to a special case involving definable partial orders, and explores related minimal group structures.

Contribution

It reduces Podewski's conjecture to a specific case involving definable well partial orders and provides new examples and classifications of minimal groups with almost linear orders.

Findings

No almost linear, minimal fields interpreting a linear order exist.

Constructed an example of an almost linear, minimal group of exponent 2.

Proved that all almost linear, minimal groups are elementary abelian of prime exponent.

Abstract

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.