Related papers: On counting totally imaginary number fields

We estimate, in a number field, the number of elements and the maximal number of linearly independent elements, with prescribed bounds on their valuations. As a by-product, we obtain new bounds for the successive minima of ideal lattices.…

Given a $2$-adic field $K$, we give formulae for the number of totally ramified quartic field extensions $L/K$ with a given discriminant valuation and Galois closure group. We use these formulae to prove a refinement of Serre's mass…

Counting number fields with prescribed Galois group is an enduring challenge in arithmetic statistics. Using the determinant method, we provide an upper bound for even groups, which is new in some cases.

The modern quantum theory is based on the assumption that quantum states are represented by elements of a complex Hilbert space. It is expected that in future quantum theory the number field will be not postulated but derived from more…

We calculate an explicit lower bound on the proportion of elliptic curves that are modular over any Galois CM field not containing $\zeta_5$. Applied to imaginary quadratic fields, this proportion is at least $2/5$. Applied to cyclotomic…

We classify all complex quadratic number fields with 2-class group of type (2,2^m) whose Hilbert 2-class fields have class groups of 2-rank equal to 2. These fields all have 2-class field tower of length 2. We still don't know examples of…

The absolute Galois group Gal$(\overline{\mathbb{Q}}/\mathbb{Q})$ of the field $\mathbb{Q}$ of rational numbers can be presented as a highly computable object, under the notion of type-2 Turing computation. We formalize such a presentation…

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…

For any given positive integer $m$ we construct certain totally positive algebraic integers $\alpha$ of a real bi-quadratic field $K$ and obtain some necessary conditions for which $m\alpha$ can not be represented as sum of integral…

Given an odd prime $\ell$ and finite set of odd primes $S_+$, we prove the existence of an imaginary quadratic field whose class number is indivisible by $\ell$ and which splits at every prime in $S_+$. Notably, we do not require that $p…

We introduce a notion of highly Kummer-faithful fields and study its relationship with the notion of Kummer-faithful fields. We also give some examples of highly Kummer-faithful fields. For example, if $k$ is a number field of finite degree…

We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every…

We produce an infinite family of imaginary quadratic fields whose ideal class groups have $3$-rank at least $2$.

We implement methods from the geometry of numbers to give explicit estimates for the number of integral ideals in a number field. We pay particular attention to minimising the effect of the degree $n$ of the number field on the error term…

This article initiates the study of topological transcendental fields $\FF$ which are subfields of the topological field $\CC$ of all complex numbers such that $\FF$ consists of only rational numbers and a nonempty set of transcendental…

In 1976, Onabe discovered that, in contrast to the Neukirch-Uchida results that were proved around the same time, a number field $K$ is not completely characterized by its absolute abelian Galois group $A_K$. The first examples of…

Let $K$ be a number field, $A/K$ be an absolutely simple abelian variety of CM type, and $\ell$ be a prime number. We give explicit bounds on the degree over $K$ of the division fields $K(A[\ell^n])$, and when $A$ is an elliptic curve we…

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that…

We consider random fields that can be represented as integrals of deterministic functions with respect to infinitely divisible random measures and show that these random fields are infinitely divisible.

The list of norm-Euclidean imaginary quadratic fields is known and finite. For each known case, we give a division algorithm that finds a remainder at distance less than the Euclidean minimum of the field.