Divisibility of Class Numbers: Enumerative Approach

Article The Gauss-Hasse hypothesis on real quadratic fields with class number one was published on January 1, 1982 in the journal Journal für die reine und angewandte Mathematik (volume 1982, issue 335).

Let k=Q(d) be a real quadratic numbefield where d > 0 is a positive square-free integer. The map d→Q(d) is a bijection from the set off all square-free integers d ≠ 0, 1 to the set of all quadratic fields Q(d)={x+yd|x,y∈Q}. Furthermore, integral basis element of algebraic integer’s ring in real quadratic fields is determined by either wd=d=[a0;a1,a2,⋯,al(d)−1,2a0¯] in the case of d ≡ 2,3(mod 4) or wd=1+d2=[a0;a1,a2,⋯,al(d)−1,2a0−1¯] in the case of d ≡ 1(mod 4) where l(d) is the period length of continued fraction expansion.The purpose of this paper is to obtain classification of some types of real quadratic fields Q(d), which include the specific form of continued fraction expansion of integral basis element wd, for which has all partial quotient elements are equal to each other and written as ξs (except the last digit of the period) for ξ positive even integer where period length is l=l(d) and d ≡ 2,3(mod 4) is a square free positive integer. Moreover, the present paper deals with determining new certain...

Explicit computation of Gross–Stark units over real quadratic fields

Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family $$ {\mathcal{K}}_n $$ , where K ∈ $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ $$ {\mathcal{K}}_n $$ with $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)].

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

In this paper we consider the relationship between real quadratic fields, their class numbers and the continued fraction expansion of related ideals, as well as the prime-producing capacity of certain canonical quadratic polynomials. This continues and extends work in [10]–[31] and is related to work in [3]–[4].

We construct some real quadratic fields whose class number is more than one from a class of sequences of positive integers. We also give a lower bound on the class number of such fields using Tatuzawa’s theorem and Yokoi invariant.

We study partitions of totally positive integers in real quadratic fields. We develop an algorithm for computing the number of partitions, prove a result about the parity of the partition function, and characterize the quadratic fields such that there exists an element with exactly 1–5, 7, and 11 partitions.

Uniform distribution and real fields

On Real Quadratic Fields whose Class Numbers are Multiples of 3.

Real quadratic fields with class numbers divisible by n

Well-rounded twists of ideal lattices from real quadratic fields

The concept of regular and irregular primes has played an important role in number theory at least since the time of Kummer. We extend this concept to the setting of arbitrary totally real number fields k o, using the values of the zeta function ζk0 at negative integers as our “higher Bernoulli numbers”. Once we have defined k 0-regular primes and the index of k 0-irregularity, we discuss how to compute these indices when k 0 is a real quadratic field. Finally, we present the results of some preliminary computations, and show that the frequency of various indices seems to agree with those predicted by a heuristic argument.

Let $K = \\mathbf{Q}(\\sqrt{m})$ be a real quadratic field, $\\mathcal{O}_K$\nits ring of integers and $G = \\operatorname{Gal}(K/\\mathbf{Q})$.\nFor $\\gamma \\in H^1(G, \\mathcal{O}_K^{\\times})$, we associate\na module $M_c/P_c$ for $\\gamma = [c]$. It is known that\n$M_c/P_c \\approx \\mathbf{Z}/\\Delta_m \\mathbf{Z}$ where $\\Delta_m = 1$\nor 2 and we will determine $\\Delta_m$.

Sums of nonvanishing integral squares in real quadratic fields