60 research outputs found
Canonical bases in excellent classes
We show that any (atomic) excellent class can be expanded with hyperimaginaries to form an (atomic) excellent class which has canonical bases. When is, in addition, of finite U-rank, then is also simple and has a full canonical bases theorem. This positive situation contrasts starkly with homogeneous model theory for example, where the eq-expansion may fail to be homogeneous. However, this paper shows that expanding an ω-stable, homogeneous class gives rise to an excellent class, which is simple if is of finite U-ran
Around Podewski's conjecture
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
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 of exponent 2, and we show that
each almost linear, minimal group is torsion.Comment: 16 page
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