scholarly journals A Property of Meta-Abelian Extensions

1961 ◽  
Vol 19 ◽  
pp. 169-187 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Abelian Group ◽  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Normal Extension ◽  
Finite Degree ◽  
Abelian Extensions ◽  
Abelian Field

Let k be an algebraic number field of finite degree, A the maximal abelian extension over k, and M a meta-abelian field over h of finite degree, that is, M/k be a normal extension over k of finite degree with an abelian group as commutator group of its Galois group.

scholarly journals Galois Group of the Maximal Abelian Extension over an Algebraic Number Field

1957 ◽  
Vol 12 ◽  
pp. 177-189 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Finite Degree ◽  
Class Field ◽  
Continuous Character

The aim of the present work is to determine the Galois group of the maximal abelian extension ΩA over an algebraic number field Ω of finite degree, which we fix once for all.Let Z be a continuous character of the Galois group of ΩA/Ω. Then, by class field theory, the character Z is also regarded as a character of the idele group of Ω. We call such Z character of Ω. For our purpose, it suffices to determine the group Xl of the characters of Ω whose orders are powers of a prime number l.

scholarly journals The Notion of Restricted Idèles with Application to Some Extension Fields

1966 ◽  
Vol 27 (1) ◽  
pp. 121-132
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Class Field Theory ◽  
Normal Extension ◽  
Finite Degree ◽  
Ground Field ◽  
Class Field ◽  
Abelian Extensions

Let k be an algebraic number field of finite degree, K be its normal extension of degree n, and ŝ be the set of those primes of K which have degree 1. Using this set s instead of the set of all primes of K, we define an s-restricted idèle of K by the same way as ordinary idèles. It is known by Bauer that the normal extension of an algebraic number field is determined by the set of all primes of the ground field which are decomposed completely in the extension field. This suggests that if we treat abelian extensions over K which are normal over k, the class field theory is expressed by means of the ŝ-restricted idèles (theorem 2). When K = k, ŝ is the set of all primes of K, and we have the ordinary class field theory.

scholarly journals The Genus Field and Genus Number in Algebraic Number Fields

1967 ◽  
Vol 29 ◽  
pp. 281-285 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Abelian Extension ◽  
Normal Extension ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Genus Field ◽  
Genus Number

Let k be an algebraic number field and K be its normal extension of finite degree. Then the genus field K* of K over k is defined as the maximal unramified extension of K which is obtained from K by composing an abelian extension over k2). We call the degree (K*: K) the genus number of K over k.

ON THE DENSITY OF DISCRIMINANTS OF ABELIAN EXTENSIONS OF A NUMBER FIELD

2010 ◽  
Vol 06 (06) ◽  
pp. 1273-1291
Author(s):  
BEHAILU MAMMO
Keyword(s):  
Asymptotic Formula ◽  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Prime Order ◽  
Algebraic Number Field ◽  
Abelian Extensions ◽  
Cyclic Groups

Let G = Cℓ × Cℓ denote the product of two cyclic groups of prime order ℓ, and let k be an algebraic number field. Let N(k, G, m) denote the number of abelian extensions K of k with Galois group G(K/k) isomorphic to G, and the relative discriminant 𝒟(K/k) of norm equal to m. In this paper, we derive an asymptotic formula for ∑m≤XN(k, G; m). This extends the result previously obtained by Datskovsky and Mammo.

scholarly journals A generalization of Hubert’s theorem 94

1991 ◽  
Vol 121 ◽  
pp. 161-169 ◽  
Author(s):  
Hiroshi Suzuki
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Group Transfer

In this paper we shall prove the following theorem conjectured by Miyake in [3] (see also Jaulent [2]).THEOREM. Let k be a finite algebraic number field and K be an unramified abelian extension of k, then all ideals belonging to at least [K: k] ideal classes of k become principal in K.Since the capitulation homomorphism is equivalently translated to a group-transfer of the galois group (see Miyake [3]), it is enough to prove the following group-theoretical verison:

scholarly journals A norm residue map for central extensions of an algebraic number field

1984 ◽  
Vol 93 ◽  
pp. 61-69 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Galois Extension ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Factor Group ◽  
Central Extensions ◽  
Norm Residue ◽  
Genus Field

Let K be a finite Galois extension of an algebraic number field k with G = Gal (K/k), and M be a Galois extension of k containing K. We denote by resp. the genus field resp. the central class field of K with respect to M/k. By definition, the field is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k. Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H-3(G, Z), and is isomorphic to H-3(G, Z) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).

scholarly journals On the unramified common divisor of discriminants of integers in a normal extension

2000 ◽  
Vol 160 ◽  
pp. 181-186
Author(s):  
Satomi Oka
Keyword(s):  
Prime Ideal ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Extension ◽  
Greatest Common Divisor ◽  
Finite Degree

AbstractLet F be an algebraic number field of a finite degree, and K be a normal extension over F of a finite degree n. Let be a prime ideal of F which is unramified in K/F, be a prime ideal of K dividing such that . Denote by δ(K/F) the greatest common divisor of discriminants of integers of K with respect to K/F. Then, divides δ(K/F) if and only if

scholarly journals On the deduction of the class field theory from the general reciprocity of power residues

2000 ◽  
Vol 160 ◽  
pp. 135-142
Author(s):  
Tomio Kubota ◽  
Satomi Oka
Keyword(s):  
Field Theory ◽  
Number Field ◽  
Algebraic Number ◽  
Quadratic Extension ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Finite Degree ◽  
Reciprocity Law ◽  
Special Cases ◽  
Cyclotomic Extension

AbstractWe denote by (A) Artin’s reciprocity law for a general abelian extension of a finite degree over an algebraic number field of a finite degree, and denote two special cases of (A) as follows: by (AC) the assertion (A) where K/F is a cyclotomic extension; by (AK) the assertion (A) where K/F is a Kummer extension. We will show that (A) is derived from (AC) and (AK) only by routine, elementarily algebraic arguments provided that n = (K : F) is odd. If n is even, then some more advanced tools like Proposition 2 are necessary. This proposition is a consequence of Hasse’s norm theorem for a quadratic extension of an algebraic number field, but weaker than the latter.

scholarly journals On the Imbedding Problem of Normal Algebraic Number Fields

1952 ◽  
Vol 4 ◽  
pp. 55-61 ◽  
Author(s):  
Eizi Inaba
Keyword(s):  
Galois Group ◽  
Finite Groups ◽  
Algebraic Number ◽  
Invariant Subgroup ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Factor Group ◽  
Normal Extension ◽  
Finite Degree ◽  
Algebraic Number Fields

Let G and H be finite groups. If a group G̅ has an invariant subgroup H̅, which is isomorphic with H, such that the factor group G̅/H̅ is isomorphic with G. then we say that G̅ is an extension of H by G. Now let G be the Galois group of a normal extension K over an algebraic number field k of finite degree. The imbedding problem concerns us with the question, under what conditions K can be imbedded in a normal extension L over k such that the Galois group of L over k is isomorphic with G̅ and K corresponds to H̅. Brauer connected this problem with the structure of algebras over k, whose splitting fields are isomorphic with K.

scholarly journals A Cohomological Investigation of the Discriminant of a Normal Algebraic Number Field

1966 ◽  
Vol 27 (1) ◽  
pp. 207-211 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Extension ◽  
Finite Degree

1. Let F be an algebraic number field of finite degree, and let K/F be a normal extension of degree n. Denote by OK the ring of all integers in K. In we proved the following:

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