Related papers: Totally real bi-quadratic fields with large P\'{o}…

For an integral domain $R$, the {\it ring of integer-valued polynomials} over $R$ consists of all polynomials $f(X) \in R[X]$ such that $f(R) \subseteq R$. An interesting case to study is when $R$ is a Dedekind domain, in particular when…

For an algebraic number field $K$, the P\'{o}lya group of $K$, denoted by $Po(K),$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of same norm. The number field $K$ is said…

The P\'{o}lya group $Po(K)$ of an algebraic number field $K$ is the subgroup of the ideal class group $Cl_{K}$ generated by the ideal classes of the products of prime ideals of the same norm. If $Po(K)$ is trivial, then the number field $K$…

The P\'olya group ${\rm Po}(K)$ of a number field $K$ is the subgroup of the ideal class group ${\rm Cl}(K)$ of $K$ generated by the classes of all the products of the prime ideals of $K$ with the same norm. Motivated by the classical "one…

In this paper we study the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ for the rings of integers in totally real biquadratic fields $K$. We continue the work of Tinkov\'a towards proving the conjecture by Kr\'asensk\'y, Ra\v{s}ka and…

For any integer $k\ge 1$, we show that there are infinitely many complex quadratic fields whose 2-class groups are cyclic of order $2^k$. The proof combines the circle method with an algebraic criterion for a complex quadratic ideal class…

The P\'{o}lya group of an algebraic number field is the subgroup generated by the ideal classes of the products of prime ideals of equal norm inside the ideal class group. Inspired by a recent work on consecutive quadratic fields with large…

The P\'olya-Ostrowski group of a Galois number field $K$, is the subgroup $Po(K)$ of the ideal class group $Cl(K)$ of $K$ generated by the classes of all the strongly ambiguous ideals of $K$. The number field $K$ is called a P\'olya field,…

We prove that for any given positive integer $\ell$ there are infinitely many imaginary quadratic fields with 2-class group of type $(2,2^\ell)$, and provide a lower bound for the number of such groups with bounded discriminant for…

Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$,…

Let $n>1$ be an odd integer. We prove that there are infinitely many imaginary quadratic fields of the form $\mathbb{Q}(\sqrt{x^2-2y^n})$ whose ideal class group has an element of order $n$. This family gives a counter example to a…

Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we…

We examine the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ of the ring of integers $\mathcal{O}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family…

Let $a\geq 1$ and $n>1$ be odd integers. For a given prime $p$, we prove under certain conditions that the class groups of imaginary quadratic fields $\mathbb{Q}(\sqrt{a^2-4p^n})$ have a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. We…

For any positive integer $n$, we show that there exists a real number field $k$ (resp. $k'$) of degree $2^n$ whose $2$-class group is isomorphic $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ such that the Galois group of the maximal…

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

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2,2,2) whose Hilbert 2-class fields are finite.

The P\'{o}lya group of an algebraic number field is a particular subgroup of the ideal class group. This article provides an overview of recent results on P\'{o}lya groups of number fields, their connection with the ring of integer-valued…

For a given positive integer $k$, we prove that there are at least $x^{1/2-o(1)}$ integers $d\leq x$ such that the real quadratic fields $\mathbb Q(\sqrt{d+1}),\dots,\mathbb Q(\sqrt{d+k})$ have class numbers essentially as large as…

We show how to construct infinite families of explicitly determined cubic number fields whose class group has a subgroup isomorphic to $(\mathbb{Z}/2)^8$ using degree $1$ del Pezzo surfaces. We illustrate the method and provide an example…