Divisibility of class numbers of imaginary quadratic fields whose discriminant has only three prime factors

We prove the existence of infinitely many imaginary quadratic fields whose discriminant has exactly three distinct prime factors and whose class group has an element of a fixed large order. The main tool we use is solving an additive problem via the circle method.     

The class numbers of real quadratic fields of discriminant 4p

For p prime, p≡3 (mod 4), we study the expansion of $\sqrt p$ into a continued fraction. In particular, we show that in the expansion $$\sqrt p = [n,\overline {l_1 ,...,l_L ,l,L_L ,...,l_1 ,2n} ]$$ l₁, ... l_L satisfy at least L/2 linear relations. We also obtain a new lower bound for the fundamental unit ε_p of the field ?( $\sqrt p$ ) for almost all p under consideration: ε_p > p³/log^1+δp for all p≥x with O(x/log^1+δx) possible exceptions (here δ>0 is an arbitrary constant), and an estimate for the mean value of the class number of ?( $\sqrt p$ ) with respect to averaging over ε_p: $$\sum\limits_{p \equiv 3 (\bmod 4), \varepsilon _p \leqslant x} {h(p) = O(x)}$$ . Bibliography: 11 titles.     

The distribution of prime ideals of imaginary quadratic fields

Let be a primitive positive definite quadratic form with integer coefficients. Then, for all there exist such that is prime and

This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.

     

Congruence properties of class numbers of quadratic fields

The authors prove that the class number of the quadratic field Q(√?g) is divisible by 3 if g is a prime of the form 27n² + 4.     

Computations of class numbers of real quadratic fields

In this paper an unconditional probabilistic algorithm to compute the class number of a real quadratic field is presented, which computes the class number in expected time . The algorithm is a random version of Shanks' algorithm. One of the main steps in algorithms to compute the class number is the approximation of . Previous algorithms with the above running time , obtain an approximation for by assuming an appropriate extension of the Riemann Hypothesis. Our algorithm finds an appoximation for without assuming the Riemann Hypothesis, by using a new technique that we call the `Random Summation Technique'. As a result, we are able to compute the regulator deterministically in expected time . However, our estimate of on the running time of our algorithm to compute the class number is not effective.

     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

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.     

Estimate of quadratic trigonometric sums with prime numbers

An estimate of the modulus of an exponential sum over primes

$S_2 \left( {\alpha ;x,1} \right) = \sum\limits_{n \leqslant x} {\Lambda \left( n \right)e\left( {\alpha \left( {n + 1} \right)^2 } \right)}$

is obtained, where α is approximated by a rational number with a large denominator.

     

Real quadratic fields with class numbers divisible by n

In this note I prove that the class number of Q(√Δ(x)) is infinitely often divisible by n, where Δ(x) = x²ⁿ + 4.     

Note on the class numbers of certain real quadratic fields

We prove that the class number of the real quadratic field is divisible byn forany integern ≥ 2 andany odd integera ≥ 3.     

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     

complex numbers;quadratic equations with a negtive discriminant

<正>One property of a real number is that its square is nonnegative.For example,there is no rea number x for which x~2=-1.To remedy this situation we introduce a number called the imaginary unit,which we denote by i and whose square is -1.Thus,     

Simultaneous indivisibility of class numbers of pairs of real quadratic fields

The Ramanujan Journal - For a square-free integer t, Byeon (Proc. Am. Math. Soc. 132:3137–3140, 2004) proved the existence of infinitely many pairs of quadratic fields $$mathbb...