scholarly journals On Greenberg’s generalized conjecture for imaginary quartic fields

2020 ◽  
pp. 1-11
Author(s):  
Naoya Takahashi
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Group Ring ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Prime Numbers ◽  
Quadratic Fields ◽  
Imaginary Quadratic Fields ◽  
Number Formula

For an algebraic number field [Formula: see text] and a prime number [Formula: see text], let [Formula: see text] be the maximal multiple [Formula: see text]-extension. Greenberg’s generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-[Formula: see text] extension of [Formula: see text] is pseudo-null over the completed group ring [Formula: see text]. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

scholarly journals p-Torsion points on elliptic curves defined over quadratic fields

1984 ◽  
Vol 96 ◽  
pp. 139-165 ◽  
Author(s):  
Fumiyuki Momose
Keyword(s):  
Elliptic Curves ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Quadratic Fields ◽  
Finite Degree ◽  
Torsion Points

Let p be a prime number and k an algebraic number field of finite degree d. Manin [14] showed that there exists an integer n = n(k,p) (≧0) which satisfies the condition

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 On the maximal abelian ℓ-extension of a finite algebraic number field with given ramification

1978 ◽  
Vol 70 ◽  
pp. 183-202 ◽  
Author(s):  
Hiroo Miki
Keyword(s):  
Natural Number ◽  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Algebraic Number Field

Let k be a finite algebraic number field and let ℓ be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively : (1) the class number of the cyclotomic Zℓ-extension of k,(2) the Galois group of the maximal abelian ℓ-extension of k with given ramification,(3) the number of independent cyclic extensions of k of degree ℓ, which can be extended to finite cyclic extensions of k of any ℓ-power degree, and(4) a certain subgroup Bk(m, S) (cf. § 2) of k×/k×)ℓm for any natural number m (see the main theorem in §3).

scholarly journals On the Ring of Integers in an Algebraic Number Field as a representation Module of Galois Group

1960 ◽  
Vol 16 ◽  
pp. 83-90 ◽  
Author(s):  
Hideo Yokoi
Keyword(s):  
Irreducible Representation ◽  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Prime Order ◽  
Algebraic Number Field ◽  
Integral Representations ◽  
Indecomposable Representation ◽  
List Type ◽  
Ring Of Integers

1. Introduction. It is known that there are only three rationally inequivalent classes of indecomposable integral representations of a cyclic group of prime order l. The representations of these classes are: (I) identical representation,(II) rationally irreducible representation of degree l – 1,(III) indecomposable representation consisting of one identical representation and one rationally irreducible representation of degree l-1 (F. E. Diederichsen [1], I. Reiner [2]).

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.

On tamely ramified pro-p-extensions over -extensions of

2013 ◽  
Vol 156 (2) ◽  
pp. 281-294
Author(s):  
TSUYOSHI ITOH ◽  
YASUSHI MIZUSAWA
Keyword(s):  
Galois Group ◽  
Prime Number ◽  
Number Field ◽  
Rational Number ◽  
Prime Numbers ◽  
Finite Set

AbstractFor an odd prime number p and a finite set S of prime numbers congruent to 1 modulo p, we consider the Galois group of the maximal pro-p-extension unramified outside S over the ${\mathbb Z}_p$-extension of the rational number field. In this paper, we classify all S such that the Galois group is a metacyclic pro-p group.

scholarly journals Unit Groups of Cyclic Extensions

1957 ◽  
Vol 12 ◽  
pp. 221-229 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Unit Group ◽  
Algebraic Number Field ◽  
Cyclic Extension ◽  
Finite Degree ◽  
Norm Group

Let Ω be an algebraic number field of finite degree, which we fix once for all, and let K be a cyclic extension over Ω such that the degree of K/Ω is a powerof a prime number l. It is obvious that the norm group NK/ΩeK of the unit group ek of K, being a subgroup of the unit group e of Ω contains the groupconsisting of all-th powersof ε∈e.

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 On some properties of group rings

1980 ◽  
Vol 29 (4) ◽  
pp. 385-392 ◽  
Author(s):  
G. Karpilovsky
Keyword(s):  
Commutative Ring ◽  
Number Field ◽  
Group Ring ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Group Rings ◽  
Arbitrary Group ◽  
Bijective Correspondence ◽  
Isomorphism Classes ◽  
Ring Of Algebraic Integers

AbstractLet Out (RG) be the set of all outer R-automorphisms of a group ring RG of arbitrary group G over a commutative ring R with 1. It is proved that there is a bijective correspondence between the set Out (RG) and a set consisting of R(G × G)-isomorphism classes of R-free R(G × G)-modules of a certain type. For the case when G is finite and R is the ring of algebraic integers of an algebraic number field the above result implies that there are only finitely many conjugacy classes of group bases in RG. A generalization of a result due to R. Sandling is also provided.

Sign in / Sign up

Export Citation Format

Share Document