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.     

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.     

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.     

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...     

Overpartitions and real quadratic fields

It is shown that counting certain differences of overpartition functions is equivalent to counting elements of a given norm in appropriate real quadratic fields.     

On real quadratic function fields

Let q be a power of an odd prime number p and K:=Fq(T) be the rational function field with a fixed indeterminate T. For P a prime of K, let be the maximal real subfield of the Pth-cyclotomic function field and its ring of integers. We prove that there exists infinitely many primes P of even degree such that the cardinal of the ideal class group is divisible by q. We prove also an analogous result for imaginary extensions.     

Class numbers of complex quadratic fields

Congruence conditions on the class numbers of complex quadratic fields have recently been studied by various investigators, including Barrucand and Cohn, Hasse, and the author. In this paper, we study the class number of Q(√ ? pq), where p ≡ q (mod 4) are distinct primes.     

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).     

On units of real quadratic fields

Let ∈ = (t + u(d)2/2 be a unit of Q((d)1/2), whose norm = 1. We investigate the properties of the factors of the number u_n which is defined by ∈ⁿ = (t_n + u_n(d)1/2)/2.     

Invariants of formally real fields and quadratic forms

Let F be a formally real field. Connections between the invariants I(F), N(F), ud(F), ũd(F), β_F(i) (i ϵ [1, l(F) - 1]) are studied, using a new invariant C(F). The possibilities for β_F are described when l(F) ≤ 5 or ud(F) ≤ 4, without considering existence.     

Indecomposable totally positive numbers in real quadratic orders

In a totally real number field, every totally positive integral number is a finite sum of (additively) indecomposable totally positive integral numbers, and up to multiplication by totally positive units, there exist only finitely many indecomposables. In the paper it is shown that in quadratic fields all these numbers can be listed in a very efficient way by using the so-called intermediate convergents of a certain quadratic irrationality. The method can be viewed as a simple extension of the standard method of calculating the fundamental unit by using continued fractions. As an application it is shown that for instance in Z|√d| a number is decomposable if its norm is >d. It is remarkable that this bound does not depend on the size of the fundamental unit.     

2-universal -lattices over real quadratic fields

Let be a real quadratic field with m a square-free positive rational integer, and be the ring of integers in F. An -lattice L on a totally positive definite quadratic space V over F is called r-universal if L represents all totally positive definite -lattices l with rank r over . We prove that there exists no 2-universal -lattice over F with rank less than 6, and there exists a 2-universal -lattice over F with rank 6 if and only if m=2, 5. Moreover there exists only one 2-universal -lattice with rank 6, up to isometry, over .