scholarly journals Central extensions and rational quadratic forms

1993 ◽  
Vol 130 ◽  
pp. 177-182 ◽  
Author(s):  
Yoshiomi Furuta ◽  
Tomio Kubota
Keyword(s):  
Central Extension ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Narrow Sense ◽  
Maximal Extension ◽  
Central Extensions ◽  
Class Field ◽  
Ray Class Field

The purpose of this paper is to characterize by means of simple quadratic forms the set of rational primes that are decomposed completely in a non-abelian central extension which is abelian over a quadratic field. More precisely, let L = Q be a bicyclic biquadratic field, and let K = Q. Denote by the ray class field mod m of K in narrow sense for a large rational integer m. Let be the maximal abelian extension over Q contained in and be the maximal extension contained in such that Gal(/L) is contained in the center of Gal(/Q). Then we shall show in Theorem 2.1 that any rational prime p not dividing d1d2m is decomposed completely in /Q if and only if p is representable by rational integers x and y such that x ≡ 1 and y ≡ 0 mod m as followswhere a, b, c are rational integers such that b2 − 4ac is equal to the discriminant of K and (a) is a norm of a representative of the ray class group of K mod m.Moreover is decomposed completely in if and only if .

scholarly journals Binary quadratic forms and ray class groups

2019 ◽  
Vol 150 (2) ◽  
pp. 695-720 ◽  
Author(s):  
Ick Sun Eum ◽  
Ja Kyung Koo ◽  
Dong Hwa Shin
Keyword(s):  
Quadratic Forms ◽  
Quadratic Field ◽  
Congruence Subgroup ◽  
Abelian Extension ◽  
Prime Ideals ◽  
Binary Quadratic Forms ◽  
Class Groups ◽  
Extended Form ◽  
Form Class ◽  
Ray Class Field

AbstractLet K be an imaginary quadratic field different from $\open{Q}(\sqrt {-1})$ and $\open{Q}(\sqrt {-3})$. For a positive integer N, let KN be the ray class field of K modulo $N {\cal O}_K$. By using the congruence subgroup ± Γ1(N) of SL2(ℤ), we construct an extended form class group whose operation is basically the Dirichlet composition, and explicitly show that this group is isomorphic to the Galois group Gal(KN/K). We also present an algorithm to find all distinct form classes and show how to multiply two form classes. As an application, we describe Gal(KNab/K) in terms of these extended form class groups for which KNab is the maximal abelian extension of K unramified outside prime ideals dividing $N{\cal O}_K$.

scholarly journals Truncated Euler Systems over Imaginary Quadratic Fields

2009 ◽  
Vol 195 ◽  
pp. 97-111
Author(s):  
Soogil Seo
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Abelian Extension ◽  
Iwasawa Theory ◽  
Quadratic Fields ◽  
Galois Module ◽  
Imaginary Quadratic Fields ◽  
Euler Systems ◽  
Graded Module ◽  
Elliptic Units

AbstractLet K be an imaginary quadratic field and let F be an abelian extension of K. It is known that the order of the class group ClF of F is equal to the order of the quotient UF/ElF of the group of global units UF by the group of elliptic units ElF of F. We introduce a filtration on UF/ElF made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.

DISTRIBUTION OF GALOIS GROUPS OF MAXIMAL UNRAMIFIED 2-EXTENSIONS OVER IMAGINARY QUADRATIC FIELDS

2018 ◽  
Vol 237 ◽  
pp. 166-187
Author(s):  
SOSUKE SASAKI
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Imaginary Quadratic Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Generalized Quaternion ◽  
Class Field Tower

Let $k$ be an imaginary quadratic field with $\operatorname{Cl}_{2}(k)\simeq V_{4}$. It is known that the length of the Hilbert $2$-class field tower is at least $2$. Gerth (On 2-class field towers for quadratic number fields with$2$-class group of type$(2,2)$, Glasgow Math. J. 40(1) (1998), 63–69) calculated the density of $k$ where the length of the tower is $1$; that is, the maximal unramified $2$-extension is a $V_{4}$-extension. In this paper, we shall extend this result for generalized quaternion, dihedral, and semidihedral extensions of small degrees.

scholarly journals On a problem of Hasse and Ramachandra

2019 ◽  
Vol 17 (1) ◽  
pp. 131-140
Author(s):  
Ja Kyung Koo ◽  
Dong Hwa Shin ◽  
Dong Sung Yoon
Keyword(s):  
Positive Integer ◽  
Quadratic Field ◽  
Imaginary Quadratic Field ◽  
Integral Ideal ◽  
Weber Function ◽  
Class Field ◽  
Ray Class Field

Abstract Let K be an imaginary quadratic field, and let 𝔣 be a nontrivial integral ideal of K. Hasse and Ramachandra asked whether the ray class field of K modulo 𝔣 can be generated by a single value of the Weber function. We completely resolve this question when 𝔣 = (N) for any positive integer N excluding 2, 3, 4 and 6.

The rational characterization of certain sets of relatively Abelian extensions

1959 ◽  
Vol 251 (998) ◽  
pp. 385-425 ◽  
Keyword(s):  
Field Theory ◽  
Galois Group ◽  
Class Group ◽  
Class Field Theory ◽  
Abelian Extension ◽  
Galois Groups ◽  
Class Field ◽  
Abelian Extensions ◽  
Rational Field

Let H be a class group— in the sense of class-field theory— in the rational field P, whose order is some power of a prime l . With H there is associated an Abelian extension K of P. The purpose of this paper is to determine in rational terms and for all fields K given in the described manner, the set T(K/P) of cyclic extensions A of K of relative degree l , which are absolutely normal. In particular we shall find the ramification laws for these fields A, and the possible extension types of a group of order l by the Galois group of K, which are realized in Galois groups of fields in T(K/P). It is fundamental to the programme outlined, that we aim at obtaining purely rational criteria of determination.

scholarly journals THE SECOND p-CLASS GROUP OF A NUMBER FIELD

2012 ◽  
Vol 08 (02) ◽  
pp. 471-505 ◽  
Author(s):  
DANIEL C. MAYER
Keyword(s):  
Automorphism Group ◽  
Galois Group ◽  
Number Field ◽  
Class Group ◽  
Quadratic Field ◽  
Quadratic Fields ◽  
Class Numbers ◽  
Green Is ◽  
Class Field ◽  
Type 3

For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group [Formula: see text] of the second Hilbert p-class field [Formula: see text] of K are determined by the p-class numbers of the unramified cyclic extensions Ni|K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field [Formula: see text] and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields Li|ℚ of absolute degree p of the dihedral fields Ni. As an application, the structure of the automorphism group [Formula: see text] of the second Hilbert 3-class field [Formula: see text] is analyzed for all quadratic fields K with discriminant -106 < D < 107 and 3-class group of type (3, 3) by computing their principalization types. The distribution of these metabelian 3-groups G on the coclass graphs [Formula: see text], 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated.

scholarly journals Gauss’s ternary form reduction and its application to a prime decomposition symbol

1985 ◽  
Vol 98 ◽  
pp. 77-86 ◽  
Author(s):  
Yoshiomi Furuta
Keyword(s):  
Diophantine Equation ◽  
Central Extension ◽  
Quadratic Forms ◽  
Quadratic Field ◽  
Square Root ◽  
Ideal Class ◽  
Explicit Method ◽  
Prime Decomposition ◽  
Ternary Form ◽  
Quadratic Diophantine Equation

We defined a prime decomposition symbol [d1 d2, p] in a previous paper [3], and characterized in [4] the set of rational primes p which are decomposed completely in a non-abelian central extension which is of degree 8 in substance. The explicit value of the symbol was determined by using a solution of certain ternary quadratic diophantine equation. The solution corresponds to a square root of an ideal class of the principal genus of a quadratic field. This is translated to a problem in classes of integral quadratic forms, namely to find a form whose duplication is a given one contained in a principal genus. An explicit method to find the form is given by Gauss in [5, Art. 286], which is due to his ternary form reduction.

The group Gal(k3(2)|k) for k = ℚ(−3,d) of type (3,3)

2016 ◽  
Vol 12 (07) ◽  
pp. 1951-1986 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Mohamed Talbi ◽  
Mohammed Talbi ◽  
Aïssa Derhem ◽  
Daniel C. Mayer
Keyword(s):  
Galois Group ◽  
Class Group ◽  
Quadratic Field ◽  
Galois Groups ◽  
Numerical Verification ◽  
Real Quadratic Field ◽  
Class Field ◽  
Type Formula ◽  
Biquadratic Fields ◽  
Real Quadratic

Let [Formula: see text] denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields [Formula: see text], having a [Formula: see text]-class group of type [Formula: see text], the possibilities for the isomorphism type of the Galois group [Formula: see text] of the second Hilbert [Formula: see text]-class field [Formula: see text] of [Formula: see text] are determined. For each coclass graph [Formula: see text], [Formula: see text], in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots [Formula: see text] of even branches of exactly one coclass tree and, in the case of even coclass [Formula: see text], additionally their siblings of depth [Formula: see text] and defect [Formula: see text], turn out to be admissible. The principalization type [Formula: see text] of [Formula: see text]-classes of [Formula: see text] in its four unramified cyclic cubic extensions [Formula: see text] is given by [Formula: see text] for [Formula: see text], and by [Formula: see text] for [Formula: see text]. The theory is underpinned by an extensive numerical verification for all [Formula: see text] fields [Formula: see text] with values of [Formula: see text] in the range [Formula: see text], which supports the assumption that all admissible vertices [Formula: see text] will actually be realized as Galois groups [Formula: see text] for certain fields [Formula: see text], asymptotically.

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