scholarly journals Kummer theory for norm algebraic tori

2005 ◽  
Vol 293 (2) ◽  
pp. 427-447 ◽  
Author(s):  
Masanari Kida
Keyword(s):  
Algebraic Tori ◽  
Kummer Theory

RAMIFICATION IN KUMMER EXTENSIONS ARISING FROM ALGEBRAIC TORI

2017 ◽  
Vol 96 (2) ◽  
pp. 196-204
Author(s):  
MASAMITSU SHIMAKURA
Keyword(s):  
Classical Theory ◽  
Multiplicative Group ◽  
Algebraic Tori ◽  
Kummer Theory ◽  
Weil Restriction

We describe the ramification in cyclic extensions arising from the Kummer theory of the Weil restriction of the multiplicative group. This generalises the classical theory of Hecke describing the ramification of Kummer extensions.

scholarly journals QUINTIC POLYNOMIALS OF HASHIMOTO–TSUNOGAI, BRUMER AND KUMMER

2009 ◽  
Vol 05 (04) ◽  
pp. 555-571 ◽  
Author(s):  
MASANARI KIDA ◽  
GUÉNAËL RENAULT ◽  
KAZUHIRO YOKOYAMA
Keyword(s):  
Isomorphism Problem ◽  
Algebraic Tori ◽  
Kummer Theory

We establish an isomorphism between the quintic cyclic polynomials discovered by Hashimoto–Tsunogai and those arising from Kummer theory for certain algebraic tori. This enables us to solve the isomorphism problem for Hashimoto–Tsunogai polynomials and also Brumer's quintic polynomials.

Kummer theory for number fields via entanglement groups

2021 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba ◽  
Sebastiano Tronto
Keyword(s):  
Number Fields ◽  
Kummer Theory

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 Corrigenda On a classification of the function fields of algebraic tori: (Nagoya Math. J. 56 (1975), 85–104)

1980 ◽  
Vol 79 ◽  
pp. 187-190 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Direct Product ◽  
Cyclic Group ◽  
Prime Divisor ◽  
Dihedral Group ◽  
Quaternion Group ◽  
Algebraic Tori ◽  
Root Of Unity ◽  
Generalized Quaternion Group ◽  
Generalized Quaternion

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

Integral structures in algebraic tori

1995 ◽  
Vol 59 (5) ◽  
pp. 881-897 ◽  
Author(s):  
V E Voskresenskii ◽  
T V Fomina
Keyword(s):  
Algebraic Tori ◽  
Integral Structures

scholarly journals Kummer theory for number fields and the reductions of algebraic numbers

2019 ◽  
Vol 15 (08) ◽  
pp. 1617-1633 ◽  
Author(s):  
Antonella Perucca ◽  
Pietro Sgobba
Keyword(s):  
Number Field ◽  
Arithmetic Progression ◽  
Direct Proof ◽  
Number Fields ◽  
Algebraic Numbers ◽  
Kummer Theory ◽  
Multiplicative Order ◽  
Higher Rank ◽  
Multiplicative Subgroup ◽  
Torsion Free

For all number fields the failure of maximality for the Kummer extensions is bounded in a very strong sense. We give a direct proof (without relying on the Bashmakov–Ribet method) of the fact that if [Formula: see text] is a finitely generated and torsion-free multiplicative subgroup of a number field [Formula: see text] having rank [Formula: see text], then the ratio between [Formula: see text] and the Kummer degree [Formula: see text] is bounded independently of [Formula: see text]. We then apply this result to generalize to higher rank a theorem of Ziegler from 2006 about the multiplicative order of the reductions of algebraic integers (the multiplicative order must be in a given arithmetic progression, and an additional Frobenius condition may be considered).

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.

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