At the $4^{th}$ chapter of Algebraic Theory of Numbers, Pierre Samuel introduces the discrete subgroups of $\mathbb{R^n}$. We have, given a discrete subgroup $H$, it is generated as a $\mathbb{Z}-$module by $r$ vectors ($r \leq n)$ which are linearly independent over $\mathbb{R}$.
Also we have $P_e$, half open parallelotope, defined as: $$P_e:=\{x \in \mathbb{R^n} | x = \sum_{i=1}^{n}\alpha_ie_i, 0\leq\alpha_i< 1 \}$$ where $e=(e_1,\dots,e_n)$ is a $\mathbb{Z}-$base for $H$.
Given a measurable subset $S$ of $\mathbb{R^n}$, $\mu(S)$ denotes the Lebesgue measure of $S$.
Now, I will state a lemma and its proof. Then I will ask few questions about the subject and the proof.
Lemma. The volume $\mu(P_e)$ is independent of the base $e$ chosen for $H$.
Proof. Let $f=(f_1, \dots,f_n)$ be another base for $H$. Then, $f_i = \sum_{i=1}^{n}\alpha_{ij}e_j$. By calculus, we know that $\mu(P_f) = |\det(\alpha_{ij})|\mu(P_e)$. The matrix $(\alpha_{ij})$ being associated with a change of base, is invertible with an inverse integer matrix, so $\det(\alpha_{ij})= \pm 1$. Thus, $\mu(P_f) = \mu(P_e)$.
So, I would like to ask:
- I can see that we have $f^T = (\alpha_{ij})e^T$ but how can we say $\mu(P_f) = |\det(\alpha_{ij})|\mu(P_e)$?
- Even though I have checked many books on algebraic number theory, they directly use lattices without introducing it right before the class number formula. How much lattice theory knowledge do I need to continue and where can I find them?
- Any advices to see the relation between modules and lattices.
Thank you.