scholarly journals On the genus field of an algebraic number field of odd prime degree

1975 ◽  
Vol 27 (2) ◽  
pp. 289-293 ◽  
Author(s):  
Makoto ISHIDA
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Prime Degree ◽  
Genus Field

On correspondence between orders and ideals with a normal basis in cyclic subfields of Q(ξp) of a prime degree l

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
Juraj Kostra
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Normal Basis ◽  
Prime Degree ◽  
One To One

AbstractLet K be a tamely ramified cyclic algebraic number field of prime degree l. In the paper one-to-one correspondence between all orders of K with a normal basis and all ideals of K with a normal basis is given.

scholarly journals On unramified cyclic extensions of degree l of algebraic number fields of degree l

1987 ◽  
Vol 107 ◽  
pp. 135-146 ◽  
Author(s):  
Yoshitaka Odai
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Preceding Paper ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Maximal Extension ◽  
Algebraic Number Fields ◽  
Cubic Field ◽  
Genus Field ◽  
Abelian Number Field

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

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 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.

scholarly journals On the rank of the first radical layer of a p-class group of an algebraic number field

1999 ◽  
Vol 156 ◽  
pp. 85-108
Author(s):  
Hiroshi Yamashita
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sylow Subgroup ◽  
Class Group ◽  
Cyclotomic Field ◽  
Artinian Ring ◽  
Algebraic Number Field ◽  
Central Extensions ◽  
Genus Field ◽  
Global And Local

Let p be a prime number. Let M be a finite Galois extension of a finite algebraic number field k. Suppose that M contains a primitive pth root of unity and that the p-Sylow subgroup of the Galois group G = Gal(M/k) is normal. Let K be the intermediate field corresponding to the p-Sylow subgroup. Let = Gal(K/k). The p-class group C of M is a module over the group ring ZpG, where Zp is the ring of p-adic integers. Let J be the Jacobson radical of ZpG. C/JC is a module over a semisimple artinian ring Fp. We study multiplicity of an irreducible representation Φ apperaring in C/JC and prove a formula giving this multiplicity partially. As application to this formula, we study a cyclotomic field M such that the minus part of C is cyclic as a ZpG-module and a CM-field M such that the plus part of C vanishes for odd p.To show the formula, we apply theory of central extensions of algebraic number field and study global and local Kummer duality between the genus group and the Kummer radical for the genus field with respect to M/K.

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