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.

Quadratic forms overR(t)

1966 ◽  
Vol 62 (2) ◽  
pp. 197-205 ◽  
Author(s):  
James T. Knight
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Class Number ◽  
Quadratic Forms ◽  
Rational Functions ◽  
Algebraic Number Field ◽  
Detailed Treatment ◽  
The Subject

AbstractA theory of quadratic forms is developed overR(t), the field of rational functions with real coefficients, instead of over an algebraic number field: some results are analogous to the classical ones, but many, in particular the non-finiteness of the class number, whose more detailed treatment will be the subject of a later paper, are not analogous.

scholarly journals On S-class number relations of algebraic tori in Galois extensions of global fields

1991 ◽  
Vol 124 ◽  
pp. 133-144 ◽  
Author(s):  
Masanori Morishita
Keyword(s):  
Class Number ◽  
Quadratic Forms ◽  
Euler Number ◽  
Number Fields ◽  
Binary Quadratic Forms ◽  
Algebraic Number Fields ◽  
Algebraic Tori ◽  
Genus Theory ◽  
Class Number Formula ◽  
Number Formula

As an interpretation and a generalization of Gauss’ genus theory on binary quadratic forms in the language of arithmetic of algebraic tori, Ono [02] established an equality between a kind of “Euler number E(K/k)” for a finite Galois extension K/k of algebraic number fields and other arithmetical invariants associated to K/k. His proof depended on his Tamagawa number formula [01] and Shyr’s formula [Sh] which follows from the analytic class number formula of a torus. Later, two direct proofs were given by Katayama [K] and Sasaki [Sa].

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 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 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.

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.

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