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 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.

Algebraic extensions in nonstandard models and Hilbert's irreducibility theorem

1988 ◽  
Vol 53 (2) ◽  
pp. 470-480 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Natural Number ◽  
Prime Number ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Algebraic Extension ◽  
Algebraic Integers ◽  
Nonstandard Models ◽  
Ring Of Algebraic Integers

LetKbe an algebraic number field andIKthe ring of algebraic integers inK. *Kand *IKdenote enlargements ofKandIKrespectively. LetxЄ *K–K. In this paper, we are concerned with algebraic extensions ofK(x)within *K. For eachxЄ *K–Kand each natural numberd, YK(x,d)is defined to be the number of algebraic extensions ofK(x)of degreedwithin *K.xЄ *K–Kis called a Hilbertian element ifYK(x,d)= 0 for alldЄ N,d> 1; in other words,K(x)has no algebraic extension within *K. In their paper [2], P. C. Gilmore and A. Robinson proved that the existence of a Hilbertian element is equivalent to Hilbert's irreducibility theorem. In a previous paper [9], we gave many Hilbertian elements of nonstandard integers explicitly, for example, for any nonstandard natural numberω, 2ωPωand 2ω(ω3+ 1) are Hilbertian elements in*Q, where pωis theωth prime number.

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 On some class number relations for Galois extensions

1987 ◽  
Vol 107 ◽  
pp. 121-133 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Quadratic Forms ◽  
Algebraic Number Field ◽  
Narrow Sense ◽  
Binary Quadratic Forms ◽  
Finite Degree ◽  
Algebraic Tori ◽  
Number Of Classes

Let k be an algebraic number field of finite degree over Q, the field of rationals, and K be an extension of finite degree over k. By the use of the class number of algebraic tori, we can introduce an arithmetical invariant E(K/k) for the extension K/k. When k = Q and K is quadratic over Q, the formula of Gauss on the genera of binary quadratic forms, i.e. the formula where = the class number of K in the narrow sense, the number of classes is a genus of the norm form of K/Q and tK = the number of distinct prime factors of the discriminant of K, may be considered as an equality between E(K/Q) and other arithmetical invariants of K.

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.

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.

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