Chebyshev's bias in dihedral and generalized quaternion Galois groups

On quaternion algebra over the composite of quadratic number fields and over some dihedral fields

Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of quaternion division algebras H_K(p,q) over some dihedral fields K.

具有四元数和二面体扩张的双二次数域的计数

Counting biquadratic number fields with quaternionic and dihedral extensions

本文研究了可嵌入四元数或二面体扩张的双二次数域的计数问题，通过用 Hilbert 符号表达逆 Galois 问题的可解性，并应用 Heath-Brown 的方法界定关联二次特征标的和，建立了有界判别式下此类数域数量的渐近公式。

关于四元数与二面体brace的数量及Hopf--Galois结构

On the number of quaternion and dihedral braces and Hopf--Galois structures

本研究证明了乘群为广义四元数群的给定阶braces数量的相关猜想，给出了二面体群情形的类似结果，并枚举了对应Galois扩张上的阿贝尔型Hopf-Galois结构。

Imaginärquadratische Einbettung von Ordnungen rationaler Quaternionenalgebren, und die nichtzyklischen endlichen Untergruppen der Bianchi-Gruppen

Let k be an imaginary quadratic number field, let F be a rational quaternion algebra and M an extension of F as a quaternion k-algebra. We are going to classify the F-orders which arise as an intersection of F with a maximal M-order; and we are going to prove that the discriminant of such an intersection determines uniquely the isomorphism type of the corresponding maximal M-order. Building on this, we are going to relate this intersection to the intersection of a second rational quaternion algebra F' in M with a second maximal M-order. This allows us to determine whether the Bianchi group ove…

Ramanujan Cayley graphs of the generalized quaternion groups and the Hardy-Littlewood conjecture

In this article, we investigate the bound of the valency of the Cayley graphs of the generalized quaternion groups which guarantees to be Ramanujan. As is the cases of the cyclic and dihedral groups in our previous studies, we show that the determination of the bound in a special setting is related to the classical Hardy-Littlewood conjecture for primes represented by a quadratic polynomial.

On quaternion algebras over some extensions of quadratic number fields

Let $p$ and $q$ be two positive primes. Let $\ell$ be an odd positive prime integer and $F$ a quadratic number field. Let $K$ be an extension of $F$ such that $K$ is a dihedral extension of $\Q$ of degree $\ell$ over $F$ or $K$ is an abelian $\ell$-extension unramified over $F$ assuming $\ell$ divides the class number of $F$. In this paper, we obtain a complete characterization of division quaternion algebras $H_{K}(p, q)$ over $K$.