scholarly journals Some Remarks to Ono’s theorem on a generalization of Gauss’ genus theory

1988 ◽  
Vol 111 ◽  
pp. 131-142 ◽  
Author(s):  
Ryuji Sasaki
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Rational Number ◽  
Galois Extension ◽  
Multiplicative Group ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Genus Theory

Let K\k be a finite Galois extension of finite algebraic number fields with Galois group g. We denote by Gm the multiplicative group defined over the rational number field Q and put.

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 On nilpotent extensions of algebraic number fields I

1992 ◽  
Vol 125 ◽  
pp. 1-14
Author(s):  
Katsuya Miyake ◽  
Hans Opolka
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Algebraic Number ◽  
Number Fields ◽  
Abelian Extension ◽  
Central Series ◽  
Schur Multiplier ◽  
Absolute Galois Group ◽  
Algebraic Number Fields ◽  
The Absolute

The lower central series of the absolute Galois group of a field is obtained by iterating the process of forming the maximal central extension of the maximal nilpotent extension of a given class, starting with the maximal abelian extension. The purpose of this paper is to give a cohomological description of this central series in case of an algebraic number field. This description is based on a result of Tate which states that the Schur multiplier of the absolute Galois group of a number field is trivial. We are in a profinite situation throughout which requires some functorial background especially for treating the dual of the Schur multiplier of a profinite group. In a future paper we plan to apply our results to construct a nilpotent reciprocity map.

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 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.

ON FINITE GALOIS STABLE SUBGROUPS OF GLn IN SOME RELATIVE EXTENSIONS OF NUMBER FIELDS

2009 ◽  
Vol 08 (04) ◽  
pp. 493-503 ◽  
Author(s):  
H.-J. BARTELS ◽  
D. A. MALININ
Keyword(s):  
Galois Group ◽  
Number Field ◽  
Galois Extension ◽  
Commutator Subgroup ◽  
Finite Extension ◽  
Number Fields ◽  
Maximal Order ◽  
Linear Groups ◽  
Stable Linear

Let K/ℚ be a finite Galois extension with maximal order [Formula: see text] and Galois group Γ. For finite Γ-stable subgroups [Formula: see text] it is known [4], that they are generated by matrices with coefficients in [Formula: see text], Kab the maximal abelian subextension of K over ℚ. This note gives a contribution to the corresponding question in the case of a relative Galois extension K/R, where R is a finite extension of the rationals ℚ. It turns out, that in this relative situation the answer to the corresponding question depends heavily on the arithmetic of the number field R, more precisely on the ramification behavior of primes in K/R. Due to the possibility of unramified extensions of R for certain number fields R there exist examples of Galois stable linear groups [Formula: see text] which are not fixed elementwise by the commutator subgroup of Gal (K/R).

scholarly journals Note on an Ordering Theorem for Subfields

1952 ◽  
Vol 4 ◽  
pp. 125-129
Author(s):  
Tadasi Nakayama
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Original Field

In a recent paper [3] Tannaka gave an interesting ordering theorem for subfields of a p-adic number field, The purpose of the present note is firstly to observe, on modifying Tannaka’s argument a little, that his restriction to those subfields over which the original field is abelian may be removed and in fact the theorem holds for arbitrary fields which are not p-adic number fields, indeed in a much refined form, and secondly to formulate a similar ordering theorem for algebraic number fields in terms of idèle-class groups in place of element groups.

The discriminant of compositum of algebraic number fields

2019 ◽  
Vol 15 (02) ◽  
pp. 353-360
Author(s):  
Sudesh K. Khanduja
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Sufficient Conditions ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Rational Numbers ◽  
Necessary And Sufficient Conditions ◽  
Algebraic Number Fields ◽  
Necessary And Sufficient

For an algebraic number field [Formula: see text], let [Formula: see text] denote the discriminant of an algebraic number field [Formula: see text]. It is well known that if [Formula: see text] are algebraic number fields with coprime discriminants, then [Formula: see text] are linearly disjoint over the field [Formula: see text] of rational numbers and [Formula: see text], [Formula: see text] being the degree of [Formula: see text] over [Formula: see text]. In this paper, we prove that the converse of this result holds in relative extensions of algebraic number fields. We also give some more necessary and sufficient conditions for the analogue of the above equality to hold for algebraic number fields [Formula: see text] linearly disjoint over [Formula: see text].

scholarly journals On the generators of S-unit groups in algebraic number fields

1991 ◽  
Vol 43 (2) ◽  
pp. 325-329 ◽  
Author(s):  
B. Brindza
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Number Fields ◽  
Finitely Generated ◽  
Geometry Of Numbers ◽  
Algebraic Number Fields ◽  
Minimal Set ◽  
Simple Argument ◽  
Small Height ◽  
Multiplicative Subgroup

Given a finitely generated multiplicative subgroup Us in a number field, we employ a simple argument from the geometry of numbers and an inequality on multiplicative dependence in number fields to obtain a minimal set of generators consisting of elements of relatively small height.

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