Around Podewski's conjecture
Tom 222 / 2013
Fundamenta Mathematicae 222 (2013), 175-193
MSC: Primary 03C60; Secondary 12L12, 20A15, 03C45.
DOI: 10.4064/fm222-2-4
Streszczenie
A long-standing conjecture of Podewski states that every minimal field is algebraically closed. Known in positive characteristic, it remains wide open in characteristic zero. We reduce Podewski's conjecture to the (partially) ordered case, and we conjecture that such fields do not exist. We prove the conjecture in case the incomparability relation is transitive (the almost linear case).
We also study minimal groups with a (partial) order, and give a complete classification of almost linear minimal groups as certain valued groups.
