scholarly journals A generalization of Hubert’s Theorem 94

1984 ◽  
Vol 96 ◽  
pp. 83-94 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Group ◽  
Algebraic Number Field ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Finite Degree ◽  
Class Field ◽  
The Absolute

Let k be an algebraic number field of finite degree. We denote the absolute class field of k by and the absolute ideal class group of k by Cl(k).

scholarly journals Note on class number factors and prime decompositions

1977 ◽  
Vol 66 ◽  
pp. 167-182 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Algebraic Number ◽  
Class Number ◽  
Galois Extension ◽  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Finite Degree ◽  
Ring Of Integers

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g, be a congruent ideal class group of K, and M be the class field over K corresponding to . Assume that M is normal over k. Then g acts on as a group of automorphisms. Donote by lg the augmentation ideal of the group ring Zg over the ring of integers Z.

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 On nilpotent factors of congruent ideal class groups of Galois extensions

1976 ◽  
Vol 62 ◽  
pp. 13-28 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Class Group ◽  
Algebraic Number Field ◽  
Augmentation Ideal ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Finite Degree ◽  
Ring Of Integers ◽  
Ideal Class Groups ◽  
The Ideal

Let K be a Galois extension of an algebraic number field k of finite degree with Galois group g. Then g acts on a congruent ideal class group of K as a group of automorphisms, when the class field M over K corresponding to is normal over K. Let Ig be the augmentation ideal of the group ring Zg over the ring of integers Z, namely Ig be the ideal of Zg generated by σ − 1, σ running over all elements of g. Then is the group of all elements aσ-1 where a and σ belong to and g respectively.

scholarly journals A PRECISE RESULT ON THE ARITHMETIC OF NON-PRINCIPAL ORDERS IN ALGEBRAIC NUMBER FIELDS

2012 ◽  
Vol 11 (05) ◽  
pp. 1250087 ◽  
Author(s):  
ANDREAS PHILIPP
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Theoretical Approach ◽  
Class Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Precise Result ◽  
Principal Order

Let R be an order in an algebraic number field. If R is a principal order, then many explicit results on its arithmetic are available. Among others, R is half-factorial if and only if the class group of R has at most two elements. Much less is known for non-principal orders. Using a new semigroup theoretical approach, we study half-factoriality and further arithmetical properties for non-principal orders in algebraic number fields.

On a Theorem of Kawamoto on Normal Bases of Rings of Integers, II

2005 ◽  
Vol 48 (4) ◽  
pp. 576-579 ◽  
Author(s):  
Humio Ichimura
Keyword(s):  
Prime Number ◽  
Number Field ◽  
Class Group ◽  
Natural Homomorphism ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Integral Basis ◽  
Rings Of Integers ◽  
Integer Ring ◽  
The Ideal

AbstractLet m = pe be a power of a prime number p. We say that a number field F satisfies the property when for any a ∈ F×, the cyclic extension F(ζm, a1/m)/F(ζm) has a normal p-integral basis. We prove that F satisfies if and only if the natural homomorphism is trivial. Here K = F(ζm), and denotes the ideal class group of F with respect to the p-integer ring of F.

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 Nonstandard Arithmetic of Polynomial Rings

1987 ◽  
Vol 105 ◽  
pp. 33-37 ◽  
Author(s):  
Masahiro Yasumoto
Keyword(s):  
Rational Function ◽  
Number Field ◽  
Algebraic Number ◽  
Algebraic Number Field ◽  
Polynomial Rings ◽  
Finite Degree

Let f(X, T1,…, Tm) be a polynomial over an algebraic number field k of finite degree. In his paper [2], T. Kojima provedTHEOREM. Let K = Q. if for every m integers t1, …, tm, there exists an r ∈ K such that f(r, t1, …, tm) =), then there exists a rational function g(T1,…,Tm) over Q such thatF(g(T1,…,Tm), T1,…,T)= 0.

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