相关论文: Class Numbers of Orders in Quartic Fields

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…

In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every…

In this paper, we construct infinitely many quadruples of real quadratic fields whose class numbers are all divisible by $3$. To the best of our knowledge, this is the first result towards the divisibility of the class numbers of certain…

In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class…

Let $K$ be an imaginary quadratic field and $\mathcal{O}$ be an order in $K$. We construct class fields associated with form class groups which are isomorphic to certain $\mathcal{O}$-ideal class groups in terms of the theory of canonical…

Using an elementary identity, we prove that for infinitely many polynomials $P(x)\in \mathbb{Z}[X]$ of fourth degree, the equation $\prod\limits_{k=1}^{n}P(k)=y^2$ has finitely many solutions in $\mathbb{Z}$. We also give an example of a…

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 prove that the Pythagoras number of the ring of integers of the compositum of all real quadratic fields is infinite. The same holds for certain infinite totally real cyclotomic fields. In contrast, we construct infinite degree totally…

We consider infinite parametric families of high degree number fields composed of quadratic fields with pure cubic, pure quartic, pure sextic fields and with the so called simplest cubic, simplest quartic fields. We explicitly describe an…

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…

For a square-free integer $t$, Byeon \cite{byeon} proved the existence of infinitely many pairs of quadratic fields $\mathbb{Q}(\sqrt{D})$ and $\mathbb{Q}(\sqrt{tD})$ with $D > 0$ such that the class numbers of all of them are indivisible…

In 2024, M. K. Ram proved that the class number of an imaginary cyclic quartic number field is never equal to a prime $p\equiv 3\pmod 4$. Here we greatly generalize this result to the case of the non-quadratic imaginary cyclic number fields…

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$.…

In this paper we study the distribution of orders of bounded discriminants in number fields. We give an asymptotic formula for the number of orders contained in the ring of integers of a quintic number field.

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…

In this paper, we investigate the 2-rank of the class group of some real cyclic quartic number fields. Precisely, we consider the case where the quadratic subfield is Q(\sqrt{l}) with l congruent to 5 modulo 8 is a prime.

Let $n$ be a squarefree positive odd integer. We will show that there exist infinitely many imaginary quadratic number fields with discriminant divisible by $n$ and-at the same time-having an element of order $n$ in the class group. We then…

Let $K$ be a totally real number field of degree $n$ over $\mathbb{Q}$, with discriminant and regulator $\Delta_K, R_K$ respectively. In this paper, using a similar method to van Woerden, we prove that the number of classes of perfect unary…

We investigate the large values of class numbers of cubic fields, showing that one can find arbitrary long sequences of "close" abelian cubic number fields with class numbers as large as possible. We also give a first step toward an…