scholarly journals On the Imbedding Problem of Normal Algebraic Number Fields

1952 ◽  
Vol 4 ◽  
pp. 55-61 ◽  
Author(s):  
Eizi Inaba
Keyword(s):  
Galois Group ◽  
Finite Groups ◽  
Algebraic Number ◽  
Invariant Subgroup ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Factor Group ◽  
Normal Extension ◽  
Finite Degree ◽  
Algebraic Number Fields

Let G and H be finite groups. If a group G̅ has an invariant subgroup H̅, which is isomorphic with H, such that the factor group G̅/H̅ is isomorphic with G. then we say that G̅ is an extension of H by G. Now let G be the Galois group of a normal extension K over an algebraic number field k of finite degree. The imbedding problem concerns us with the question, under what conditions K can be imbedded in a normal extension L over k such that the Galois group of L over k is isomorphic with G̅ and K corresponds to H̅. Brauer connected this problem with the structure of algebras over k, whose splitting fields are isomorphic with K.

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 On central extensions of a Galois extension of algebraic number fields

1984 ◽  
Vol 93 ◽  
pp. 133-148 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Galois Extension ◽  
Central Extension ◽  
Algebraic Closure ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Central Extensions

Let k be an algebraic number field of finite degree, and K a finite Galois extension of k. A central extension L of K/k is an algebraic number field which contains K and is normal over k, and whose Galois group over K is contained in the center of the Galois group Gal(L/k). We denote the maximal abelian extensions of k and K in the algebraic closure of k by kab and Kab respectively, and the maximal central extension of K/k by MCK/k. Then we have Kab⊃MCK/k⊃kab·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.

scholarly journals Leopoldt kernels and central extensions of algebraic number fields

1990 ◽  
Vol 120 ◽  
pp. 67-76 ◽  
Author(s):  
Katsuya Miyake
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Unit Group ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Prime Divisors ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Central Extensions ◽  
The Real

Let k be an algebraic number field of finite degree, and p be a fixed rational prime. We denote the set of all the non-Archimedian prime divisors of k by S0(k) and the set of all the real Archimedian ones by (k). Put and S = S0 ∪ S∞, and define a subgroup of the unit group (k) of k by

scholarly journals On the integer ring of the compositum of algebraic number fields

1980 ◽  
Vol 77 ◽  
pp. 25-31 ◽  
Author(s):  
Toshitaka Kataoka
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Finite Extension ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Finite Degree ◽  
Algebraic Number Fields ◽  
Integer Ring

Let k be an algebraic number field of finite degree. For a finite extension L/k we denote by L/k the different of L/k, and by L the integer ring of L. Let K1 and K2 be finite extensions of k.

scholarly journals Computing subfields in algebraic number fields

1990 ◽  
Vol 49 (3) ◽  
pp. 434-448 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Minimal Polynomial ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Local Information ◽  
Algebraic Number Fields

AbstractLet K:= Q(α) be an algebraic number field which is given by specifying the minimal polynomial f(X) for α over Q. We describe a procedure for finding the subfields L of K by constructing pairs (w(X), g(X)) of polynomials over Q such that L= Q(w(α)) and g(X) is the minimal polynomial for w(α). The construction uses local information obtained from the Frobenius-Chebotarev theorem about the Galois group Gal(f), and computations over p-adic extensions of Q.

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.

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 unramified cyclic extensions of degree l of algebraic number fields of degree l

1987 ◽  
Vol 107 ◽  
pp. 135-146 ◽  
Author(s):  
Yoshitaka Odai
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Preceding Paper ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Maximal Extension ◽  
Algebraic Number Fields ◽  
Cubic Field ◽  
Genus Field ◽  
Abelian Number Field

Let I be an odd prime number and let K be an algebraic number field of degree I. Let M denote the genus field of K, i.e., the maximal extension of K which is a composite of an absolute abelian number field with K and is unramified at all the finite primes of K. In [4] Ishida has explicitly constructed M. Therefore it is of some interest to investigate unramified cyclic extensions of K of degree l, which are not contained in M. In the preceding paper [6] we have obtained some results about this problem in the case that K is a pure cubic field. The purpose of this paper is to extend those results.

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