On solving relative norm equations in algebraic number fields

Let ${\Bbb Q} \subseteq{\cal E} \subseteq{\cal F}$ be algebraic number fields and $M\subset {\cal F}$ a free $o_{{\cal E} }$-module. We prove a theorem which enables us to determine whether a given relative norm equation of the form $|\mathop {N_{{\cal F} /{\cal E} }^{}}(\eta )| = |\theta |$ has any solutions $\eta \in M$ at all and, if so, to compute a complete set of nonassociate solutions. Finally we formulate an algorithm using this theorem, consider its algebraic complexity and give some examples.