The Computation of Gal$(k^\infty/k)$ for some Complex Quadratic Number Fields $k$
Elliot Benjamin, Franz Lemmermeyer, Chip Snyder
TL;DR
The paper determines the Galois group $\operatorname{Gal}(k^\infty/k)$ for a family of imaginary quadratic fields $k$ with discriminant $d_k=-4 p q q'$ under precise congruence and Legendre-symbol constraints, establishing that $\operatorname{Gal}(k^\infty/k)\cong \Gamma_{n,m}$ where $2^n=h_2(k)/4$ and $2^m=h_2(-p)$. It derives an explicit metabelian presentation for $\Gamma_{n,m}$ and proves the necessary genus-field and intermediate-field arithmetic (via genus theory and Kuroda formulas) to fix the structure of the $2$-class groups of related fields, including $\operatorname{Cl}_2(k^1)\simeq (2,2^m)$ and $\operatorname{Cl}_2(k_{gen})\simeq (2^n,2^m)$. A detailed group-theoretic analysis of transfers and capitulation ties the arithmetic data to the group presentation, confirming the unique realization of $\Gamma_{n,m}$ (with $\varepsilon=1$) and clarifying when low-parameter variants occur or are unrealizable. The paper also extends these results to select $n=1$ or $m=1$ cases, providing explicit realizability criteria and giving concrete computational examples that validate the theoretical framework. Overall, the work advances explicit Galois-group determinations for $2$-class towers in a notable family of imaginary quadratic fields and connects genus-theoretic and transfer techniques to precise group presentations.
Abstract
We determine the Galois group of the 2-class field tower for two particular families of imaginary quadratic number fields $k$ with $2$-class field tower of length $2$.