Related papers: Class Numbers of Orders in Quartic Fields

The number F(h) of imaginary quadratic fields with a given class number h is of classical interest: Gauss' class number problem asks for a determination of those fields counted by F(h). The unconditional computation of F(h) for h up to 100…

We construct a new infinite family of pairs of imaginary cyclic fields of degree $(p-1)/2$ explicitly with both class numbers divisible by a given prime number $p$. For the proof, we use the fundamental unit of $\mathbb Q(\sqrt{p})$,…

The distribution of $\alpha p$ modulo one, where $p$ runs over the rational primes and $\alpha$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $\nu>0$ one can establish the infinitude…

This paper presents an adaptation of recently developed algorithms for quadratic forms over number fields in arXiv:1304.0708 to global function fields of odd characteristics. First, we present algorithm for checking if a given…

Orders in number fields provide natural examples of lattices. We ask: what can the successive minima of lattices arising from orders in number fields be? Given an order $\mathcal{O}$ of absolute discriminant $\Delta$ in a degree $n$ number…

It is known that infinitely many number fields and function fields of any degree $m$ have class number divisible by a given integer $n$. However, significantly less is known about the indivisibility of class numbers of such fields. While…

The prime geodesic theorem for cycles in Bruhat-Tits buildings is applied to unit groups of division algebras to derive new asymptotic assertion on class numbers of orders in imaginary quadratic fields.

We recall the notion of nearest integer continued fractions over the Euclidean imaginary quadratic fields $K$ and characterize the "badly approximable" numbers, ($z$ such that there is a $C(z)>0$ with $|z-p/q|\geq C/|q|^2$ for all $p/q\in…

We show that for $100\%$ of the odd, squarefree integers $n > 0$ the $4$-rank of $\text{Cl}(\mathbb{Q}(i, \sqrt{n}))$ is equal to $\omega_3(n) - 1$, where $\omega_3$ is the number of prime divisors of $n$ that are $3$ modulo $4$.

In this paper, we calculate the unit groups and the $2$-class numbers of the fields $ \mathbb{K}= \mathbb{Q}(\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$ and $ \mathbb{L}= \mathbb{Q}( \sqrt{-1},\sqrt{2}, \sqrt{p_1}, \sqrt{p_2})$, where $p_1$ and…

This paper presents fundamental algorithms for the computational theory of quadratic forms over number fields. In the first part of the paper, we present algorithms for checking if a given non-degenerate quadratic form over a fixed number…

We give a precise description of how the class group of a number field measures the failure of unique factorization in its ring of integers. Specifically, following ideas of Kummer, we determine the structure of all irreducible…

The simple symplectic triple systems over the real numbers are classified up to isomorphism, and linear models of all of them are provided. Besides the split cases, one for each complex simple Lie algebra, there are two kinds of non-split…

If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number…

We prove Kitaoka's conjecture for all totally real number fields of degree 4 -- namely, there is no positive definite classical quadratic form in three variables which is universal. To achieve this, we study the fields (often without…

We introduce an algorithm that computes explicit class fields of an imaginary quadratic field $K$ for a given modulus $\mathfrak{f}\subset\mathcal{O}_K$ more efficiently than the use of their classical counterparts. Therein, we prove the…

We generalize the lemmas of Thomas Kretschmer to arbitrary number fields, and apply them with a 2-descent argument to obtain bounds for families of elliptic curves over certain imaginary quadratic number fields with class number 1. One such…

Using the complex Klein-Gordon field as a model, we quantize the quaternionic scalar field in the real Hilbert space. The lagrangian formulation has accordingly been obtained, as well as the hamiltonian formulation, and the energy and…

Given a number field, it is an important question in algorithmic number theory to determine all its subfields. If the search is restricted to abelian subfields, one can try to determine them by using class field theory. For this, it is…

We show that the ring of integers of $\mathbb{Q}^{\text{tr}}$ is existentially definable in the ring of integers of $\mathbb{Q}^{\text{tr}}(i)$, where $\mathbb{Q}^{\text{tr}}$ denotes the field of all totally real numbers. This implies that…