I was trying to solve the following problem from Lang's book:
a) Let $G$ be a finite abelian group. Prove that there exists an abelian extension of $\mathbb{Q}$ whose Galois group is $G$.
b) Let $k$ be a finite extension of $\mathbb{Q} $, and let $G$ be a finite abelian group. Prove that there exist infinitely many abelian extensions of $k$ whose Galois group is $G$.
Remark: The part a) I solved using Dirichlet's theorem about prime number in arithmetic progressions and structure of $\mathbb{Z_p}^{\times}$.
I have tried to solve the part b) but ran into some difficulties. Eventually I find the solution in this topic. However, I cannot ask the question there since the author of the answer probably not using MSE anymore. Moreover, one of the moments of the solutions seems to quite weird. I'll attach it below:
We can write $G= \prod_{i=1}^{k} Z_{n_i}$. Now using the fact that there are infinitely many primes $p$ of form $(n_i-1)|p$ for all $i$, we get infinitely many tuples of primes $S_j=(p_{j,1},p_{j,2},..,p_{j,k})$ such that $n_i-1|(p_{j,i}-1)$ for all $1\leq i\leq k$. Now we can make a infinite set $I$ such that for every distinct pair $i,j\in I$ we've $S_i\cap S_j=\mathbb{Q}$. Now, there is $E_j \subset \mathbb{Q}(\omega_{\prod p_{j,i}})$ such that $Gal(E_{j}/\mathbb{Q})=G$.
To do this you need to use the Galois correspondence and the fact that $Gal(\mathbb{Q}(w_n)/Q)$ is Abelian for all $n$ so, any intermediate extension is normal and hence Galois over $Q$. Now note that $E_i \cap E_j=\mathbb{Q}$ for any distinct $i,j \in I$.
Now $Gal(KE_i/K)=Gal(E_i/K\cap E_i)$ for all $i\in I$. Now since $Q \subset E_i \cap K \subset K$ and $[K:Q] <\infty$ so exist a infinite subset $I'$ of $I$ such that $E_i\cap K=E_j\cap K$ for any $i,j \in I'$. Bug then $E_i\cap K= (E_i\cap K)\cap (E_j\cap K)=K\cap (E_i\cap E_j)=K \cap \mathbb{Q}=\mathbb{Q}$ for any $i\neq j \in I'$.
So, then for any $i\in I'$ we've $Gal(KE_i/K)=Gal(E_i/\mathbb{Q})=G$. Now as $KE_{i}\cap KE_j=K$ and $I'$ is infinite so we're done.
My question is the following:
1) Everything looks good in this solution except one moment: How did he deduce that $KE_i \cap KE_j=K$ for all $i,j\in I'$ such that $i\neq j$?
I would be very grateful if somebody can clarify this moment to me, please?
Since I have spent yesterday a whole day but I was not able to understand this moment!