Quadratic Fields with Special Class Groups

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...

Special units and ideal class groups of extensions of imaginary quadratic fields

C∗-algebras Associated with Real Multiplication

Noncommutative tori with real multiplication are the irrational rotation algebras that have special equivalence bimodules. Y. Manin proposed the use of noncommutative tori with real multiplication as a geometric framework for the study of abelian class field theory of real quadratic fields. In this paper, we consider the Cuntz-Pimsner algebras constructed by special equivalence bimodules of irr...

Quadratic Fields with Cyclic 2-class Groups

For any integer k ≥ 1, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order 2. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class to be a square. In memory of David Hayes.

Exponents of Class Groups of Quadratic Function Fields over Finite Fields

We find a lower bound on the number of imaginary quadratic extensions of the function field Fq(T ) whose class groups have an element of a fixed order. More precisely, let q ≥ 5 be a power of an odd prime and let g be a fixed positive integer ≥ 3. There are q 1 2+ 1 g ) polynomials D ∈ Fq[T ] with deg(D) ≤ ` such that the class groups of the quadratic extensions Fq(T, √ D) have an element of or...