scholarly journals The 4-rank of the tame kernel versus the 4-rank of the narrow class group in quadratic number fields

2000 ◽  
Vol 96 (2) ◽  
pp. 155-165 ◽  
Author(s):  
Qin Yue ◽  
Keqin Feng
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Fields ◽  
Tame Kernel ◽  
Narrow Class

On the 8-rank of narrow class groups of ℚ(−4pq), ℚ(−8pq), and ℚ(8pq)

2018 ◽  
Vol 14 (08) ◽  
pp. 2165-2193 ◽  
Author(s):  
Djordjo Z. Milovic
Keyword(s):  
Lower Bounds ◽  
Class Group ◽  
Number Fields ◽  
Prime Numbers ◽  
Quadratic Residue ◽  
Quadratic Number ◽  
Class Groups ◽  
Residue Symbol ◽  
Quadratic Number Fields ◽  
Narrow Class

Let [Formula: see text]. We study the [Formula: see text]-part of the narrow class group in thin families of quadratic number fields of the form [Formula: see text], where [Formula: see text] are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order [Formula: see text]. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields.

ELEMENTS OF ORDER FOUR IN THE NARROW CLASS GROUP OF REAL QUADRATIC FIELDS

2015 ◽  
Vol 100 (1) ◽  
pp. 21-32
Author(s):  
ELLIOT BENJAMIN ◽  
C. SNYDER
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Ideal Class Group ◽  
Quadratic Number ◽  
Gaussian Integers ◽  
Real Quadratic Fields ◽  
Order Four ◽  
Real Quadratic ◽  
Narrow Class

Using the elements of order four in the narrow ideal class group, we construct generators of the maximal elementary $2$-class group of real quadratic number fields with even discriminant which is a sum of two squares and with fundamental unit of positive norm. We then give a characterization of when two of these generators are equal in the narrow sense in terms of norms of Gaussian integers.

scholarly journals Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

scholarly journals RELATIONS IN THE 2-CLASS GROUP OF QUADRATIC NUMBER FIELDS

2012 ◽  
Vol 93 (1-2) ◽  
pp. 115-120 ◽  
Author(s):  
F. LEMMERMEYER
Keyword(s):  
Class Group ◽  
Number Fields ◽  
Quadratic Number ◽  
Quadratic Number Fields

AbstractWe construct a family of ideals representing ideal classes of order two in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.

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