Let $K, L$ be number fields (i.e. subfields of $\mathbb C$ of finite degree over $\mathbb Q$) of degree $m, n$ over $\mathbb Q$ respectively and assume $[KL:\mathbb Q]=nm$.
Consider $KL$ to be the smallest field containing both $K$ and $L$.
Let $\mathbb A$ be the ring of algebraic integers in $\mathbb C$, i.e. the set of all complex numbers $\alpha$ s.t. $\exists f\in\mathbb Z[X]$ monic, $f(\alpha)=0$.
We define $R:=\mathbb A\cap K$ the ring of algebraic integers of $K$. Similarly $S:=\mathbb A\cap L$ and $T:=\mathbb A\cap KL$.
We know form the theory that $R\simeq\mathbb Z^{m},\; S\simeq\mathbb Z^{n}$, we have that $T\simeq\mathbb Z^{nm}$.
Now denote $RS$ as the smallest ring containing both $R$ and $S$. Clear that $RS\subseteq T$.
Let now $\{\alpha_1,\dots,\alpha_m\}$ be an integral basis for $R$ (i.e. a basis for $R$ over $\mathbb Z$), and $\{\beta_1,\dots,\beta_n\}$ an integral basis for $S$.
Then $\{\alpha_i\beta_j\}$ are an integral basis of the ring $RS$.
Hence they are $nm$ linearly indipendent elements over $\mathbb Z$ which stay also in $KL$. But we recall that by hypotesis $[KL:\mathbb Q]=nm$ hence $\{\alpha_i\beta_j\}$ form a basis of $KL$ over $\mathbb Q$, i.e. for every element $x\in KL$ there exists $x_{i,j}\in\mathbb Q$ s.t. $x=\sum_{i,j}x_{i,j}\alpha_i\beta_j$.
And till here is clear. Then the book says that every $x\in T:=\mathbb A\cap KL$ can be written as $$ x=\sum_{i,j}\frac{x_{i,j}}{r}\alpha_i\beta_j $$
where $r,x_{i,j}\in\mathbb Z$ with $\operatorname{gcd}(r,\operatorname{gcd}(x_{i,j}))=1$. WHY it stay also in $\mathbb A$?
I thought that I can rewrite it as $$ rx=\sum_{i,j}x_{i,j}\alpha_i\beta_j $$ And it's clear that $\sum_{i,j}x_{i,j}\alpha_i\beta_j\in\mathbb A$, but here I'm stuck. Can someone explain why an element of $T$ could be written as above? Thanks all
This is fairly straightforward, just note that $\{\alpha_i\beta_j\}_{i,j}$ is a $\Bbb Q$-basis for $KL$, so clearly all integers are rational sums of them. That we can choose the $x_i$ to suit the given conditions, for if not, we could just pick reduced representatives. It's not claiming any special properties about the $x_{i,j}$ it's just saying some integers, which is trivial since $T\subseteq KL$ which is a $\Bbb Q$-vector space. In particular, you know since
$$\Bbb Q\otimes_{\Bbb Z}T=KL$$
that $T$ contains a rational basis for $KL$ which is an integral basis for $T$. So if you choose such a basis, $\{v_k\}_{k=1}^{mn}$, there is a change of basis matrix from $\{v_k\}_k$ to $\{\alpha_i\beta_j\}_{i,j}$ and this can give rise explicitly to the coefficients in question (if the need ever arose).