Submission date: 26 January 2012.

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 torsion.

Mathematics Subject Classification: Primary 03C60, Secondary 12L12, 20A15, 03C45

Keywords and phrases: minimal field, minimal group

Full text arXiv 1201.5709: pdf, ps.