Let ε be a quartic algebraic unit. We give necessary and sufficient conditions for (i) the quartic number field K = ℚ(ε) to contain an imaginary quadratic subfield, and (ii) for the ring of algebraic integers of K to be equal to ℤ[ε]. We also prove that the class number of such K's goes to infinity effectively with the discriminant of K.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-13,
author = {St\'ephane R. Louboutin},
title = {Some quartic number fields containing an imaginary quadratic subfield},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {139-148},
zbl = {1250.11092},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-13}
}
Stéphane R. Louboutin. Some quartic number fields containing an imaginary quadratic subfield. Colloquium Mathematicae, Tome 122 (2011) pp. 139-148. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-1-13/