Differents, discriminants and Steinitz classes

Let L | K be an extension of algebraic number fields. By a theorem of Hecke, the ideal class of the different L | K is a square in Cl L . Whereas the class of the discriminant L | K =N L | K ( L | K ) is the square of the Steinitz class s L | K in Cl K , which vanishes if and only if the ring R L of integers in L is a free module over R K , there is no distinguished square root for the different in general. In an attempt to give some interpretation for Hecke's theorem, we examine when there is r ∈Cl L such that r 2 =[ L | K ] and N L | K ( r )= s L | K (‘Steinitz root’). In this case, every ideal of L in the class r −1 is a free R K ‐module.