Normal relative integral bases for quartic fields over quadratic subfields

Abstract

Let L be a quartic number field with a quadratic subfield K. In 1986 Kawamoto gave a necessary and sufficient condition for L to have a normal relative integral basis (NRIB) over K. In this paper the authors explicitly construct a NRIB for L/K when such exists using their previous work on relative integral bases. The special cases when L is bicyclic, cyclic and pure are examined in detail.