Galois p-extensions of number fields

1.设p为素数。局部域的最大pro-p-扩张的伽罗瓦群的结构是众所周知的;它要么是自由pro-p-群，要么是Demushkin群。自由亲p-群由Nielsen-Schreier公式刻画，它描述了开子群的生成元的最小数目。对于Demushkin群也有类似的结果。在本研究中，我们引入了一个类似于Nielsen-Schreier公式的条件，并研究了满足该条件的支撑群的基本性质。我们将我们的考虑应用于有限素数集外非分歧数域的最大pro-p-扩张的Galois群，并证明了，在数域和素数p上的一些温和的限制下，这个Galois群不满足我们的条件，除非它是自由的pro-p或Demushkin。2.我们以前已经证明，最大pro-p-的Galois群，有限素数集外非分歧数域的扩张只有平凡中心，除非它是交换的，假设Leopoldt猜想。在这项研究中，我们观察到我们的讨论亲p-集团的中心和Murasugi的工作中心的链接群之间的显着的相似之处。作为应用，我们改进了以前的结果.

1.Let p be a prime. The structure of the Galois group of the maximal pro-p-extension of a local field is well known ; it is either a free pro-p-group or a Demushkin group. Free pro-p-groups are characterized by the Nielsen-Schreier formula, which describes the minimal number of generators of open subgroups. An analogous result is also known for Demushkin groups. In this research, we introduced a condition analogous to the Nielsen-Schreier formula, and investigated basic properties of prop-groups satisfying the condition. We applied our consideration to the Galois group of the maximal pro-p-extension of a number field unramified outside a finite set of primes, and showed that, under some mild restrictions on the number field and the prime p, this Galois group does not satisfy our condition, unless it is free pro-p or Demushkin.2.We have previously shown that the Galois group of the maximal pro-p-extension of a number field unramified outside a finite set of primes has trivial center unless it is abelian, assuming the Leopoldt conjecture. In this research, we observed remarkable similarities between our discussion on the center of pro-p-groups and Murasugi's work on the center of link groups. As an application, we obtained an improvement of our previous result.