Let and be algebraic number fields with , and having integer rings and , respectively. Suppose is an ideal such that is principal. Show that is principal in .
We will let and denote the ideal class groups of and , respectively. For brevity, if is a subset, then we will denote by the ideal generated by in and by the ideal generated by in .
We claim that if , then . To see this, suppose we have and such that . Then . Now if , then each is in , and so has the form . So . Conversely, , and so .
The mapping then induces a well-defined mapping given by . Moreover, since , is a group homomorphism.
In particular, if in , then in , as desired.