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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

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    I'm not quite sure what you're asking. By that point Marcus has surely shown that the group $I_L$ of non-zero fractional ideals of $L$ [maybe it would be better to write $S$, but everyone does this] is free on the set of prime ideals of $S$. So to give a homomorphism $I_L \to I_K$ you just say where the primes go, and that's what you've done. So I think your question is whether we can pass to a homomorphism $G(S) \to G(R)$, i.e., whether the image of a principal fractional ideal is principal, which is not obvious from this definition. Do I have that right?2012-06-22
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    Well hes not defined fractional ideals yet, but has shown that the ideals of R factor uniquely into prime ideals, and in a previous exercise, we prove that principal ideals will stay principal under this map2012-06-22
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    I'll try to take a look at the book later.2012-06-23

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