Integral Bases for Quartic Fields with Quadratic Subfields
Let L be a quartic number field with quadratic subfield Q([formula]). Then L = Q([formula], [formula]), where a + b[formula] is not a square in Q([formula]) and where a, b, and c may be taken to be integers with both c and (a, b) squarefree. The discriminant of L, as well as an integral basis for L, is determined explicitly in terms of congruences involving a, b, and c. These results unify the existing results in the literature for quartic fields which are pure, bicyclic, cyclic, or dihedral, and complete the incomplete results in the literature for dihedral quartic fields. It is also shown that for each squarefree integer c there are infinitely many non- pure, dihedral quartic fields L with a power basis.