相关论文: Class numbers of Imaginary bicyclic biquadratic nu…

We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $k =Q(\sqrt{2pq}, i)$, where $i=\sqrt{-1}$ and $p\equiv -q\equiv1 \pmod 4$ are different primes. For each…

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…

In this paper we give an elementary proof of results on the structure of 4-class groups of real quadratic number fields originally due to A. Scholz. In a second (and independent) section we strengthen C. Maire's result that the 2-class…

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…

Here we study algebraic function fields K, give necessary and sufficient condition for the ideal class group $H(K)$ of any real quadratic function field $K$ to have a cyclic subgroup of order $n$, and obtain eight series of such fields $K$,…

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…

A number field is said to be a CM-number field if it is a totally imaginary quadratic extension of a totally real number field. We define a totally imaginary number field to be of CM-type if it contains a CM-subfield, and of TR-type if it…

Let $D<0$ be a fundamental discriminant and denote by $E(D)$ the exponent of the ideal class group $\text{Cl}(D)$ of $K={\mathbb Q}(\sqrt{D})$. Under the assumption that no Siegel zeros exist we compute all such $D$ with $E(D)$ is a divisor…

In this paper, we investigate the unit groups, the $2$-class groups, the $2$-class field towers and the structures of the second $2$-class groups of some multiquadratic number fields of degree $8$ and $16$.

We construct an infinite family of imaginary bicyclic biquadratic number fields $k$ with the 2-ranks of their 2-class groups are $\geq3$, whose strongly ambiguous classes of $k/Q(i)$ capitulate in the absolute genus field $k^{(*)}$, which…

We obtain criteria for the class number of certain Richaud-Degert type real quadratic fields to be 3. We also treat a couple of families of real quadratic fields of Richaud-Degert type that were not considered earlier, and obtain similar…

The investigation of the ideal class group $Cl_K$ of an algebraic number field $K$ is one of the key subjects of inquiry in algebraic number theory since it encodes a lot of arithmetic information about K. There is a considerable amount of…

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.

We construct parameterized families of imaginary (resp. real) quadratic fields whose class groups have $n$-rank at least $2$.

For an algebraic number field $K$ with ring of integers $\mathcal{O}_{K}$, an important subgroup of the ideal class group $Cl_{K}$ is the {\it P\'{o}lya group}, denoted by $Po(K)$, which measures the failure of the $\mathcal{O}_{K}$-module…

As a consequence of their work, Bruce C. Berndt and Ronald J. Evans in 1977 and Larry Joel Goldstein and Michael Razar in 1976 obtained a formula for the square of the class number of an imaginary quadratic number field in terms of Dedekind…

For any odd prime $p,$ we construct an infinite family of pairs of imaginary quadratic fields $\mathbb{Q}(\sqrt{d}),\mathbb{Q}(\sqrt{d+1})$ whose class numbers are both divisible by $p.$ One of our theorems settles Iizuka's conjecture for…

In this paper we study the structure of the $3-$part of the ideal class group of a certain family of real cyclotomic fields with $3-$class number exactly $9$ and conductor equal to the product of two distinct odd primes. We employ known…

Let $d$ be an odd square-free integer, $m\geq 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta_{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic…

We investigate improvements to the algorithm for the computation of ideal class groups described by Jacobson in the imaginary quadratic case. These improvements rely on the large prime strategy and a new method for performing the linear…