Definable V-topologies, Henselianity and NIP
Definable V-topologies, Henselianity and NIP
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可定义的 V 拓扑、Henselianity 和 NIP
DOI：
10.1142/s0219061320500087
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发表时间：
2019
期刊：
影响因子：
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通讯作者：
Franziska Jahnke
中科院分区：
文献类型：
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作者：
Yatir Halevi;Assaf Hasson;Franziska Jahnke
我们开始研究可定义的[Formula：see text]-拓扑，并证明在[Formula：see text]-Henselian NIP域上至多存在一个这样的[Formula：see text]-拓扑。等价地，我们证明了如果[Formula：see text]是一个具有[Formula：see text] henselian（分别为[Formula：see text]-henselian）的双值NIP域，则[Formula：see text]和[Formula：see text]是可比的（分别为依赖的）。因此，NIP域的Shelah猜想蕴涵了NIP域的Henselianity猜想。进一步证明了后一个猜想对于任何具有dp-极小剩余域的域都成立。我们的结论表明，谢拉的猜想是等价的声明，任何NIP领域不包含在代数封闭的有限领域是[公式：见文字]-henselian。
We initiate the study of definable [Formula: see text]-topologies and show that there is at most one such [Formula: see text]-topology on a [Formula: see text]-henselian NIP field. Equivalently, we show that if [Formula: see text] is a bi-valued NIP field with [Formula: see text] henselian (respectively, [Formula: see text]-henselian), then [Formula: see text] and [Formula: see text] are comparable (respectively, dependent). As a consequence, Shelah’s conjecture for NIP fields implies the henselianity conjecture for NIP fields. Furthermore, the latter conjecture is proved for any field admitting a henselian valuation with a dp-minimal residue field. We conclude by showing that Shelah’s conjecture is equivalent to the statement that any NIP field not contained in the algebraic closure of a finite field is [Formula: see text]-henselian.