Galois groups and complete domains
Galois groups and complete domains
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伽罗瓦群和完备域
DOI：
10.1007/bf02785586
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发表时间：
1999
影响因子：
1
通讯作者：
Tamara R. Lefcourt
中科院分区：
文献类型：
--
作者：
Tamara R. Lefcourt
抽象考虑一个域
$$\hat R$$
对于非零素理想来说是完全的。本文证明了关于这类环的两个Galois理论结果。利用Grothendieck存在定理，我们证明了每一个有限群都是
$$\hat R\left[ x \right]$$
.这推广了大卫Harbater谁证明了结果的情况下，理想是最大的和域是正常的结果。因此，我们推断，如果
$$\hat R$$
是关于非零素理想完备的Noether整环，则每个有限群作为Galois群出现在
$$\hat R$$
.这证明了摩西·贾登（Moshe Jarden）提出的一个猜想的诺特情形。
AbstractConsider a domain
$$\hat R$$
that is complete with respect to a non-zero prime ideal. This paper proves two Galois-theoretic results about such rings. Using Grothendieck’s Existence Theorem we prove that every finite group occurs as the Galois group of a Galois extension of
$$\hat R\left[ x \right]$$
. This generalizes results of David Harbater who proved the result in the case where the ideal is maximal and the domain is normal. As a consequence, we deduce that if
$$\hat R$$
is a Noetherian domain that is complete with respect to a non-zero prime ideal, then every finite group occurs as a Galois group over
$$\hat R$$
. This proves the Noetherian case of a conjecture posed by Moshe Jarden.