关于Podewski猜想
Around Podewski's conjecture
摘要：
Podewski的一个长期存在的猜想指出，每个极小域都是代数闭的。瓦格纳在正特性的场中证明了这一点，但在零特性的情况下，这一点仍然是开放的。我们减少Podewski的猜想的情况下，领域有一个可定义的（在纯领域的结构），以及偏序与无限链，我们猜想，这样的领域不存在。然后，我们通过证明不存在以特定方式解释线性序的最小域来支持这个猜想;在我们的术语中，不存在几乎线性的最小域。另一方面，给出了指数为2的几乎线性极小群$(M,<,+,0)$的例子，证明了几乎线性极小群是素指数的初等交换群.另一方面，我们给出了一个例子，几乎线性，最小群$(M,<,+,0)$的指数为2，我们表明，每个几乎线性，最小群是挠。
英文摘要：
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.
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