Finding the volume of
Let K be a number field, denotes the number of real embeddings and denotes so that . Define the ring homomorphism, canonical imbedding of K, as
We only consider the 'first' complex embeddings since the rest are conjugate of the previous ones.
So, there is this lemma that I do not know how to start, what to think:
Lemma.If M is a free -module of K of rank n, and if is a -base of M, then is a lattice in , whose volume is .
The determinant on the right hand side is as far as I know. However, I appreciate any help to understand the proof of this lemma.
Edit: The lemma does not say about anything about , but it is a free -module of rank n. So the lemma is more general, is a specific case but I wrote the title as this way. You can either take a -base for and continue or work on any free -module of rank n.