Caliber number of real quadratic fields

We obtain lower bound of caliber number of real quadratic field using splitting primes in K. We find all real quadratic fields of caliber number 1 and find all real quadratic fields of caliber number 2 if d is not 5 modulo 8. In both cases, we don't rely on the assumption on ζK(1/2).     

Class groups of quadratic fields of 3-rank at least 2: Effective bounds

In this paper, we give parametric families of both real and complex quadratic number fields whose class group has 3-rank at least 2. As a consequence, we obtain that for all large positive real numbers x, the number of both real and complex quadratic fields whose class group has 3-rank at least 2 and absolute value of the discriminant ?x is >cx1/3, where c is some positive constant.     

Class number 2 problem for certain real quadratic fields of Richaud-Degert type

In this paper we will apply Biró's method in [A. Biró, Yokoi's conjecture, Acta Arith. 106 (2003) 85-104; A. Biró, Chowla's conjecture, Acta Arith. 107 (2003) 179-194] to class number 2 problem of real quadratic fields of Richaud-Degert type and will show that there are exactly 4 real quadratic fields of the form with class number 2, where n²+1 is a even square free integer.     

Densities for some real quadratic fields with infinite Hilbert 2-class field towers

Let K be a real quadratic field with 2-class rank equal to 4 or 5 and 4-class rank equal to 3. This paper computes density information for such fields to have infinite Hilbert 2-class field towers.     

Normal integral bases in quadratic and cyclic cubic extensions of quadratic fields

Let K be a number field and let G be a finite abelian group. We call K a Hilbert-Speiser field of type G if, and only if, every tamely ramified normal extension L/K with Galois group isomorphic to G has a normal integral basis. Now let C₂ and C₃ denote the cyclic groups of order 2 and 3, respectively. Firstly, we show that among all imaginary quadratic fields, there are exactly three Hilbert-Speiser fields of type $C_{2}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-1, -3, -7\}$. Secondly, we give some necessary and sufficient conditions for a real quadratic field $K = \mathbb{Q}(\sqrt {m})$ to be a Hilbert-Speiser field of type C₂. These conditions are in terms of the congruence class of m modulo 4 or 8, the fundamental unit of K, and the class number of K. Finally, we show that among all quadratic number fields, there are exactly eight Hilbert-Speiser fields of type $C_{3}: \mathbb{Q}(\sqrt {m})$, where $m \in \{-11,-3, -2, 2, 5, 17, 41, 89\}$.Received: 2 April 2002     

A density result for some imaginary quadratic fields with infinite Hilbert 2-class field towers

This paper shows that a positive proportion of the imaginary quadratic fields with 2-class rank equal to 3 have 4-class rank equal to zero and infinite Hilbert 2-class field towers. Received: 14 January 2003     

A note on the 3-rank of quadratic fields

The difference between the 3-rank of the ideal class group of an imaginary quadratic field and that of the associated real quadratic field is equal to 0 or 1. In this note, we give an infinite family of examples in each case.Received: 9 September 2002     

On class numbers of quadratic extensions¶over function fields

We consider class numbers of quadratic extensions over a fixed function field. We will show that there exist infinitely many quadratic extensions which have class numbers not being divisible by 3 and satisfy prescribed ramification conditions. Received: 24 October 1997 / Revised version: 26 February 1998     

On the unit group and the class number¶of certain composita of two real quadratic fields

The purpose of this paper is to exhibit a new family of real bicyclic biquadratic fields K for which we can write the Hasse unit index of the group generated by the units of the three quadratic subfields in the unit group E _K of K. As a byproduct, one can explicitly relate the class number of K with the product of the class numbers of the three quadratic subfields. Received: 25 July 2000 / Revised version: 12 December 2000     

Reflection theorem for divisor class groups of relative quadratic function fields

We present the reflection theorem for divisor class groups of relative quadratic function fields. Let K be a global function field with constant field Fq. Let L₁ be a quadratic geometric extension of K and let L₂ be its twist by the quadratic constant field extension of K. We show that for every odd integer m that divides q+1 the divisor class groups of L₁ and L₂ have the same m-rank.     

On the Ono invariants of imaginary quadratic number fields

J. Cohen, J. Sonn, F. Sairaiji and K. Shimizu proved that there are only finitely many imaginary quadratic number fields K whose Ono invariants OnoK are equal to their class numbers hK. Assuming a Restricted Riemann Hypothesis, namely that the Dedekind zeta functions of imaginary quadratic number fields K have no Siegel zeros, we determine all these K's. There are 114 such K's. We also prove that we are missing at most one such K. M. Ishibashi proved that if OnoK is large enough compared with hK, then the ideal class groups of K is cyclic. We give a short proof and a precision of Ishibashi's result. We prove that there are only finitely many imaginary quadratic number fields K satisfying Ishibashi's sufficient condition. Assuming our Restricted Riemann Hypothesis, we prove that the absolute values dK of their discriminants are less than 2.3⋅¹⁰⁹. We determine all these K's with dK?¹⁰⁶. There are 76 such K's. We prove that there is at most one such K with dK?1.8⋅¹⁰¹¹.     

The ideal class groups of dihedral extensions over imaginary quadratic fields and the special values of the Artin L-function

We study the relation between the minus part of the p-class subgroup of a dihedral extension over an imaginary quadratic field and the special value of the Artin L-function at 0.     

Annihilating polynomials of excellent quadratic forms

If φ is an excellent form, then it is possible to use the dimensions of the higher complements of φ to obtain an annihilating polynomial of φ of low degree. The main result of this paper is the construction of such a polynomial with the help of methods from the theory of generic splitting of quadratic forms. Received: 23 April 2007     

Selmer groups and quadratic reciprocity

In this article we study the 2-Selmer groups of number fieldsF as well as some related groups, and present connections to the quadratic reciprocity law inF.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

A quadratic field which is Euclidean but not norm-Euclidean

The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integersR is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal ideal domains which are not norm-Euclidean, there should exist examples of rings of algebraic integers which are Euclidean but not norm-Euclidean. In this paper, we give the first example for quadratic fields, the ring of integers of .     

On quadratic homogeneous quasi-translations

In this paper, we describe the structure of quadratic homogeneous quasi-translations, and we prove that, in dimension 9 and less, such a map is linearly triangularizable, or equivalently a quadratic homogeneous nice derivation is linearly conjugate to a triangular derivation.     

Primes in quadratic progressions on average

In this paper, we establish a theorem on the distribution of primes in quadratic progressions on average.     

On the norm theorem for semisingular quadratic forms

The aim of this paper is to prove some results concerning the norm theorem for semisingular quadratic forms, i.e., those which are neither nonsingular nor totally singular. More precisely, we will give necessary conditions in order that an irreducible polynomial, possibly in more than one variable, is a norm ofa semisingular quadratic form, and we prove that our conditions are sufficient if the polynomial is given by a quadratic form which represents 1. As a consequence, we extend the Cassels-Pflster subform theorem to the case of semisingular quadratic forms.     

Reduction of matrices over orders of imaginary quadratic fields

A special decomposition (called the near standard form) of (1,2)-matrices over a ring is introduced and a method for a reduction of such matrices is explained. This can be applied for a detection of elementary second order matrices among invertible second order matrices. The tool is used in detail over orders of imaginary quadratic fields, where an algorithm, a number of properties and examples are presented.