The 4-class Group of Real Quadratic Number Fields
نویسنده
چکیده
In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire’s result that the 2-class field tower of a real quadratic number field is infinite if its ideal class group has 4-rank ≥ 4, using a technique due to F. Hajir.
منابع مشابه
On the real quadratic fields with certain continued fraction expansions and fundamental units
The purpose of this paper is to investigate the real quadratic number fields $Q(sqrt{d})$ which contain the specific form of the continued fractions expansions of integral basis element where $dequiv 2,3( mod 4)$ is a square free positive integer. Besides, the present paper deals with determining the fundamental unit$$epsilon _{d}=left(t_d+u_dsqrt{d}right) 2left.right > 1$$and $n_d$ and $m_d...
متن کاملClass Numbers of Quadratic Fields Determined by Solvability of Diophantine Equations
In the literature there has been considerable attention given to the exploration of relationships between certain diophantine equations and class numbers of quadratic fields. In this paper we provide criteria for the insolvability of certain diophantine equations. This result is then used to determine when related real quadratic fields have class number bigger than 1. Moreover, based on criteri...
متن کاملComputing the Hilbert class field of real quadratic fields
Using the units appearing in Stark’s conjectures on the values of L-functions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, so that k = Q( √ dk), and let ω denote an algebraic integer such that the ring of integers of k is Ok := Z+ ωZ. An important invariant of ...
متن کاملThe Gauss Class-Number Problems
In Articles 303 and 304 of his 1801 Disquisitiones Arithmeticae [Gau86], Gauss put forward several conjectures that continue to occupy us to this day. Gauss stated his conjectures in the language of binary quadratic forms (of even discriminant only, a complication that was later dispensed with). Since Dedekind’s time, these conjectures have been phrased in the language of quadratic fields. This...
متن کاملAsymptotically Fast Discrete Logarithms in Quadratic Number Fields
This article presents algorithms for computing discrete logarithms in class groups of quadratic number elds. In the case of imaginary quadratic elds, the algorithm is based on methods applied by Hafner and McCurley HM89] to determine the structure of the class group of imaginary quadratic elds. In the case of real quadratic elds, the algorithm of Buchmann Buc89] for computation of class group a...
متن کامل