On some properties of Galois groups of unramified extensions

Abstract

Let k be an algebraic number field of finite degree and k 1 be the maximal cyclotomic extension ofk. Let Q Lk and Lk be the maximal unramified Galois extension and the maximal unramified abelian extension of k 1 respectively. We shall give some remarks on the Galois groups Gal( Q Lk=k1), Gal(Lk=k1) and Gal(Q Lk=k). One of the remarks is concerned with non-solvable quotients of Gal( Q Lk=k1) when k is the rationals, which strengthens our previous result. Introduction Let k be an algebraic number field of finite degree in a fixed algebrai c closure and n denote a primitiven-th root of unity (n 1). Let k1 be the maximal cyclotomic extension ofk, i.e., the field obtained by adjoining to k all n (n 1). Let Q Lk and Lk be the maximal unramified Galois extension and the maximal un ramified abelian extension ofk 1 respectively. By the maximality, Q Lk and Lk are both Galois extensions of k. According to the analogy between finite algebraic number fiel ds and function fields of one variable over finite constant fields, adjoining all n to a finite algebraic number field is one of the substitutes of extending the finite constan t field of the function field to its algebraic closure. Therefore, the Galois group Gal( Q Lk=k1) may be regarded as an analogue of the algebraic fundamental group of a proper sm ooth geometrically connected curve over the algebraic closure of a finite field. In this article, we shall give some remarks on the Galois grou ps Gal(Q Lk=k1), Gal(Lk=k1) and Gal(Q Lk=k). It is known that the algebraic fundamental group of a smooth g eometrically connected curve over an algebraically closed constant field has t e following property (P) except for some special cases (cf. e.g. Tamagawa [8]). Every subgroup with finite index is centerfree. (P) This is one of the properties of algebraic fundamental group s of “anabelian” algebraic varieties (cf. e.g. Ihara–Nakamura [4]). Our first rem ark is that the Galois group 2010 Mathematics Subject Classification. 11R18, 11R23.