Why is this a basis over $\mathbb{Z}$

Question

I am reading the proof of Theorem 12 in Chapter 2 in Daniel A. Marcus' Number Fields (2nd ed.), which is on p. 25. But in the second sentence, it said that "Then the $mn$ products $\alpha_i\beta_j$ form a basis for $RS$ over $\mathbb{Z}$, and also for $KL$ over $\mathbb{Q}$ (why?)." Here $K$ and $L$ are number fields, $R$ and $S$ their rings of integers, $\{\alpha_1,\ldots,\alpha_m\}$ an integral basis for $R$, and $\{\beta_1,\ldots,\beta_n\}$ an integral basis for $S$. Moreover it is assumed that $[KL:\mathbb{Q}]=mn$. Although I see that the $mn$ products $\alpha_i\beta_j$ form a generating set for $RS$ over $\mathbb{Z}$, I do not see why they are linearly independent. Please help me.

Answer 1

The $mn\ $ $\alpha_i \beta_j$ form a generating set for $KL$ over $\mathbb{Q}$ and from $[KL:\mathbb{Q}]=mn$ it follows that they must be a basis, which implies that they are independent over $\mathbb{Q}$.