Integral bases for quartic fields with quadratic subfields
Any quartic number field \(L\) with quadratic subfield \(\mathbb{Q} (\sqrt {c})\) can be expressed in the form \(L= \mathbb{Q} (\sqrt {c}, \sqrt {a+b \sqrt {c}})\), where \(a+ b\sqrt {c}\) is not a square in \(\mathbb{Q} (\sqrt {d})\) and \(a\), \(b\) and \(c\) may be taken to be integers with both \(c\) and \((a,b)\) square-free. In this paper, the authors first assert that the discriminant formula and integral basis for \(L\) obtained by themselves in terms of congruences involving \(a\), \(b\) and \(c\) [Carleton Math. Res. Ser. No. 4 (1991)] unifies already known results on the discriminant and integral basis for quartic fields which are pure, cyclic or bicyclic as corollaries. Next, they give a necessary and sufficient condition for \(L\) to be pure quartic, and using these results they prove the fact that there exist infinitely many non-pure, dihedral quartic fields with a power basis.
scientific article; zbMATH DE number 749192