关于广义Grunwald-Wang问题中的缺陷
On the defect in the generalized Grunwald--Wang problem
AI总结：
研究广义Grunwald-Wang定理中特殊情况下障碍群的有限性和有界性，通过有理函数域和仿射直线点的离散赋值构造反例，证明该群可能无限且阶无界。
摘要：
经典的Grunwald-Wang定理断言，除非我们处于所谓特殊情况，否则数域在有限个完备化处的局部循环伽罗瓦扩张可以被全局循环扩张逼近。在特殊情况下，障碍由一个2阶群度量。长期以来人们知道Grunwald-Wang定理可以推广到非常一般的赋值域背景。因此自然要问：在特殊情况下，障碍是否总是由一个有限群度量？如果是，这个群的阶是否与所考虑的位置数量无关？我们证明，一般情况下这两个问题的答案都是否定的，即使对于有理函数域和来自仿射直线点的离散赋值也是如此。这与Q或Q_p上函数域的算术有一些有趣的联系。
英文摘要：
The classical Grunwald--Wang theorem asserts that, unless we are in the so-called special case, local cyclic Galois extensions at finitely many completions of a number field can be approximated by a global cyclic extension. In the special case the obstruction is measured by a group of order 2. It has been known for a long time that the Grunwald--Wang theorem extends to a very general context of valued fields. Therefore it is natural to ask whether in the special case the obstruction is always measured by a finite group and if so, is the order of this group bounded independently of the number of places under consideration. We show that the answer to both questions is negative in general, already for rational function fields and discrete valuations coming from points of the affine line. This has some interesting links to the arithmetic of function fields over Q or Q_p.
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