We prove that a valued field of positive characteristic p that has only finitely many distinct Artin–Schreier extensions (which is a property of infinite NTP₂ fields) is dense in its perfect hull. As a consequence, it is a deeply ramified field and has p-divisible value group and perfect residue field. Further, we prove a partial analogue for valued fields of mixed characteristic and observe an open problem about 1-units in this setting. Finally, we fill a gap that occurred in a proof in an earlier paper in which we first introduced a classification of Artin–Schreier defect extensions.

中文翻译：

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具有有限多个素数缺陷扩展的值域

我们证明了一个具有积极特征的有价值的领域p只有有限多个不同的 Artin-Schreier 扩展（这是无限 NTP ₂场的属性）在其完美的外壳中是密集的。因此，它是一个分支很深的领域，并且具有p- 可分值群和完美残差域。此外，我们证明了混合特征值域的部分类比，并在此设置中观察到一个关于 1 单位的开放问题。最后，我们填补了在早期论文中的证明中出现的空白，在该论文中，我们首先介绍了 Artin-Schreier 缺陷扩展的分类。