论文标题
在完成期间，估值扩展到有理功能领域
Extensions of valuations to rational function fields over completions
论文作者
论文摘要
给定一个有价值的字段$（k，v）$及其完成$（\ widehat {k}，v）$，我们研究了$ v $ to $ v $至$ \ widehat {k}（x）$的所有可能扩展名的集合。我们表明，任何此类扩展名与基础子扩展$（k（x）| k，v）$密切相关。这些扩展之间的连接是通过最小对，关键多项式，伪-Cauchy序列和隐式恒定场来研究的。结果，我们获得了$（\ widehat {k}，v）$的强大分支理论属性。我们还提供了$（k（x），v）$的必要条件，以$（\ widehat {k}（x），v）$致密。
Given a valued field $(K,v)$ and its completion $(\widehat{K},v)$, we study the set of all possible extensions of $v$ to $\widehat{K}(X)$. We show that any such extension is closely connected with the underlying subextension $(K(X)|K,v)$. The connections between these extensions are studied via minimal pairs, key polynomials, pseudo-Cauchy sequences and implicit constant fields. As a consequence, we obtain strong ramification theoretic properties of $(\widehat{K},v)$. We also give necessary and sufficient conditions for $(K(X),v)$ to be dense in $(\widehat{K}(X),v)$.
