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 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.

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.

Cyclotomic Splitting Fields

1982 ◽  
Vol 25 (2) ◽  
pp. 222-229 ◽  
Author(s):  
R. A. Mollin
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Abelian Extension ◽  
Brauer Group ◽  
Division Algebras ◽  
Necessary And Sufficient ◽  
Cyclotomic Extension ◽  
Maximal Subfield

AbstractLet D be a division algebra whose class [D] is in B(K), the Brauer group of an algebraic number field K. If [D⊗KL] is the trivial class in B(L), then we say that L is a splitting field for D or L splits D. The splitting fields in D of smallest dimension are the maximal subfields of D. Although there are infinitely many maximal subfields of D which are cyclic extensions of K; from the perspective of the Schur Subgroup S(K) of B(K) the natural splitting fields are the cyclotomic ones. In (Cyclotomic Splitting Fields, Proc. Amer. Math. Soc. 25 (1970), 630-633) there are errors which have led to the main result of this paper, namely to provide necessary and sufficient conditions for (D) in S(K) to have a maximal subfield which is a cyclic cyclotomic extension of K, a finite abelian extension of Q. A similar result is provided for quaternion division algebras in B(K).

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 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 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.

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.

Sign in / Sign up

Export Citation Format

Share Document