Some quartic number fields containing an imaginary quadratic subfield
Volume 122 / 2011
Colloquium Mathematicum 122 (2011), 139-148
MSC: Primary 11R16; Secondary 11R27, 11R29.
DOI: 10.4064/cm122-1-13
Abstract
Let $\varepsilon$ be a quartic algebraic unit. We give necessary and sufficient conditions for (i) the quartic number field $K ={\mathbb Q}(\varepsilon)$ to contain an imaginary quadratic subfield, and (ii) for the ring of algebraic integers of $K$ to be equal to ${\mathbb Z}[\varepsilon]$. We also prove that the class number of such $K$'s goes to infinity effectively with the discriminant of $K$.
