Norm homomorphism between ideal class groups I have been working through Number Fields by D.A Marcus, and I'm stuck and need a hint, the question is in chapter 3 question 16 which goes as follows:
Let $K,L$ be number fields and $K \subset L$,  where $R,S$ are the rings of integers of K and L resp. Then denote $G(R)$, $G(S)$ for the ideal class groups of $R,S$,
I need to show that there is a homomorphism $G(S) \rightarrow G(R)$ which sends any ideal I in a given class C to the class containing $N^{L}_{K}(I)$, the thing is I'm not quite sure how to show this homomorphism sends the identity to the identity.
Here for a prime ideal $Q$ of $S$,  $N^{L}_{K}(Q)=P^{f(Q|P)}$, $P$ the prime ideal lying under $Q$ and $f(Q|P)$ the inertia degree.
Thank you 
 A: The trick for this exercise is that you need to look at exercises 14 and 15. Let us define a homomorphism $f : G(S) \to G(R)$ that sends $[I] \mapsto [N_{L/K}(I)]$. We need to check that this is well-defined and is a group homomorphism. 
Suppose that there is another ideal $J$ such that $I \sim J$. Then there are $\alpha,\beta \in S$ such that $\alpha I = \beta J$. The trick now is we can say that 
$$(\alpha)I = (\beta)J.$$
Taking ideal norms (as defined in exercise 15) we get that $$N_{L/K}\left((\alpha) \right)  N_{L/K}(I) = N_{L/K}\left( (\beta)\right) N_{L/K}(J).$$
Now by exercise 15(b) we get that
$$\begin{array}{ccc} N_{L/K}\left((\alpha) \right)   &=& \left( N_{L/K} (\alpha) \right) \\ N_{L/K}\left((\beta) \right) &=&  \left( N_{L/K} (\beta) \right).\end{array}$$
Hence 
$$\left(N_{L/K}(\alpha)\right) N_{L/K}(I) = \left(N_{L/K}(\beta)\right) N_{L/K} (J)$$
which by the same trick we used in the beginning means that
$$N_{L/K}(\alpha)N_{L/K}(I) = N_{L/K}(\beta) N_{L/K} (J),$$
hence $[N_{L/K}(I) ] = [ N_{L/K} (J)].$
By 15(c) again we see that $f$ takes the identity class to the identity class. By multiplicativity of the norm we conclude finally that $f$ is a well - defined group homomorphism from $G(S)$ to $G(R)$.
