相关论文: Class numbers of Imaginary bicyclic biquadratic nu…

Let $k\geq 3$ and $n\geq 3$ be odd integers, and let $m\geq 0$ be any integer. For a prime number $\ell$, we prove that the class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{\ell^{2m}-2k^n})$ is either divisible by $n$ or by a…

It is shown that the class number for negative discriminant $D$ can be expressed in terms of the base $B$ expansions of reduced fractions $\frac{x}{|D|}$, where $B$ is an integer prime to $D$. This result is then formulated to obtain…

Let $d$ be a square free integer and $L_d:=\mathbb{Q}(\zeta_{8},\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, $\mathrm{Cl}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.

We prove an asymptotic formula for class numbers of totlally imaginary quartic number fields, ie for number fields of degree 4 over Q with only complex embeddings. After previous work for real quadratic fields (Sarnak) and complex cubic…

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that…

We consider the class numbers of imaginary quadratic extensions $F(\sqrt{-p})$, for certain primes $p$, of totally real quadratic fields $F$ which have class number one. Using seminal work of Shintani, we obtain two elementary class number…

In this article, the standard correspondence between the ideal class group of a quadratic number field and the equivalence classes of binary quadratic forms of given discriminant is generalized to any base number field of narrow class…

We investigate the number ${\Cal F}(h)$ of imaginary quadratic fields with class number $h$. We establish an asymptotic formula for the average value of ${\Cal F}(h)$. We also establish a modest non-trivial upper bound for ${\Cal F}(h)$ and…

We construct a family of ideals representing ideal classes of order 2 in quadratic number fields and show that relations between their ideal classes are governed by certain cyclic quartic extensions of the rationals.

We adapt a known technique for searching for ideal classes of arbitrary order and then apply it to three families of number fields. We show that a family of cyclic sextic number fields has infinitely many fields in it that contain a…

Let $k \geq 1$ be a cube-free integer with $k \equiv 1 \pmod {9}$ and $\gcd(k, 7\cdot 571)=1$. In this paper, we prove the existence of infinitely many triples of imaginary quadratic fields $\mathbb{Q}(\sqrt{d})$, $\mathbb{Q}(\sqrt{d+1})$…

Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$.…

Let $h_{(m,k)}$ be the class number of $\mathbb{Q}(\sqrt{1-2m^k}).$ We prove that for any odd natural number $k,$ there exists $m_0$ such that $k \mid h_{(m,k)}$ for all odd $m > m_0.$ We also prove that for any odd $m \geq 3,$ $k \mid…

We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all…

We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.

Let n be an odd number and F an imaginary quadratic field with odd discriminant. We show that there exists infinitely many cubic fields K such that the class number of K is divisible by n and the Galois closure of K contains F.

In this paper, we give a formula for the proper class number of a binary quadratic polynomial assuming that the conductor ideal is sufficiently divisible at dyadic places. This allows us to study the growth of the proper class numbers of…

The goal is to obtain an asymptotic formula for the number of quadratic extensions with bounded discriminant of a some quadratic number field with odd class number. This extends an already known result for Q.

Here we initiate a program to study relationships between finite groups and arithmetic-geometric invariants in a systematic way. To do this we first introduce a notion of optimal module for a finite group in the setting of holomorphic mock…

In this paper, we obtain an asymptotic formula for the number of imaginary quadratic fields with prime discriminant and class number up to $H$, as $H\to \infty$. Previously, such an asymptotic was only known under the assumption of the…