Galois p-extensions of number fields

点击翻译按钮获取中文摘要

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.