arXiv subjects
Publications and source records attributed to Mohamed Mahmoud Chems-Eddin.
Let $d$ be an odd square-free integer and $ζ_8$ a primitive $8$-th root of unity. The purpose of this paper is to investigate the rank of the $2$-class group of the fields $L_d=\mathbb{Q}(ζ_8,\sqrt{d})$.
Let $d$ be a square free integer and $L_d:=\mathbb{Q}(ζ_{8},\sqrt{d})$. In the present work we determine all the fields $L_d$ such that the $2$-class group, $\mathrm{Cl}_2(L_d)$, of $L_d$ is of type $(2,4)$ or $(2,2,2)$.
Let $p\equiv -q \equiv 5\pmod 8$ be two prime integers. In this paper, we investigate the unit groups of the fields $ L_1 =\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q}, \sqrt{-1} )$ and $ L_1^+=\mathbb{Q}(\sqrt 2, \sqrt{p}, \sqrt{q} )$. Furthermore , we give the second $ 2$-class groups of the subextensions of $L_1$ as well the $2$-class groups of the fields $ L_n =\mathbb{Q}( \sqrt{p}, \sqrt{q}, ζ_{2^{n+2}} )$ and their maximal real subfelds.
Let $n\geq 3$ be an integer and $d$ an odd square-free integer. We shall compute the rank of the $2$-class group of $L_{n,d}:=\mathbb{Q}(ζ_{2^n},\sqrt{d})$, when all the prime divisors of $d$ are congruent to $\pm 3\pmod 8$ or $9\pmod{16}$.
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 $16$.