Related papers: On Class Numbers of Pure Quartic fields

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 $2$,…

For fixed $q\in\{3,7,11,19, 43,67,163\}$, we consider the density of primes $p$ congruent to $1$ modulo $4$ such that the class group of the number field $\mathbb{Q}(\sqrt{-qp})$ has order divisible by $16$. We show that this density is…

It is well known that for any prime $p\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{p})$ and $\mathbb{Q}(\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a…

We determine the $3$-class groups of $\mathbb{Q}(\sqrt[3]{p})$ and $K=\mathbb{Q}(\sqrt[3]{p},\sqrt{-3})$ when $p\equiv 4,7\bmod 9$ is a prime and $3$ is a cubic modulo $p$. This confirms a conjecture made by Barrucand-Cohn, and proves the…

The density of primes $p$ such that the class number $h$ of $\mathbb{Q}(\sqrt{-p})$ is divisible by $2^k$ is conjectured to be $2^{-k}$ for all positive integers $k$. The conjecture is true for $1\leq k\leq 3$ but still open for $k\geq 4$.…

We use a variant of Vinogradov's method to show that the density of the set of prime numbers $p\equiv -1\bmod~4$ for which the class group of the imaginary quadratic number field $\mathbb{Q}(\sqrt{-8p})$ has an element of order $16$ is…

We use Vinogradov's method to prove equidistribution of a spin symbol governing the $16$-rank of class groups of quadratic number fields $\mathbb{Q}(\sqrt{-2p})$, where $p \equiv 1 \bmod 4$ is a prime.

I present Franz Lemmermeyer's proof that if p is a prime congruent to 9 mod 16 then the class number of Q(p^(1/4)) is congruent to 2 mod 4.

Let $p\equiv 1\pmod{8}$ and $q\equiv7\pmod 8$ be two prime numbers. The purpose of this paper is to compute the unit group of the fields $\KK=\QQ(\sqrt 2, \sqrt{p}, \sqrt{q} )$ and give their $2$-class numbers.

Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and $k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$ is a prime integer and $d$ an odd square-free integer coprime to $p$. In this work,…

For a prime $p\equiv 3$ $(\text{mod }4)$, let $h(-8p)$ and $h(8p)$ be the class numbers of $\mathbb{Q}(\sqrt{-2p})$ and $\mathbb{Q}(\sqrt{2p})$, respectively. Let $\Psi(\xi)$ be the Hirzebruch sum of a quadratic irrational $\xi$. We show…

We consider the problem of determining whether a given prime p is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not…

Given a random real quadratic field from $\{ \mathbb{Q}(\sqrt{p}\,) ~|~ p \text{ primes} \}$, the conjectural probability $\mathbb{P}(h=q)$ that it has class number $q$ is given for all positive odd integers $q$. Some related conjectures of…

Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $\delta \in \{1,2\}$ with $\delta=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we…

Let $p\equiv 1\,(\mathrm{mod}\,3)$ be a prime and denote by $\zeta_3$ a primitive third root of unity. Recently, Lemmermeyer presented a conjecture about $3$-class groups of pure cubic fields $L=\mathbb{Q}(\sqrt[3]{p})$ and of their normal…

The main purpose of this paper is to find all the prime numbers p for which whenever we add to p an odd square less than p we obtain a number which has at most two different prime factors. We solve completely the cases $p\equiv 1,3,5 \pmod…

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…

For any distinct two primes $p_1\equiv p_2\equiv 3$ $(\text{mod }4)$, let $h(-p_1)$, $h(-p_2)$ and $h(p_1p_2)$ be the class numbers of the quadratic fields $\mathbb{Q}(\sqrt{-p_1})$, $\mathbb{Q}(\sqrt{-p_2})$ and…

Let $p$ be an odd prime number. In this article, we study the number of quadratic residues and non-residues modulo $p$ which are multiples of $2$ or $3$ or $4$ and lying in the interval $[1, p-1]$, by applying the Dirichlet's class number…

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…