$\text{Gal}(E_{1}*...*E_{n}/K)$ is cyclic of degree $p_{1}*...*p_{n}$ for compositum $E_{1}*...*E_{n}$ of intermediate fields with $[E_{i}:K]=p_{i}$

Question

I would like to know how to prove the following

Let $E_{i}$ be intermediate fields of a field extension $L|K$, with $E_{i}|K$ finite galois $i=1,...n$. Assume that the degree $[E_{i}:K]$ is a prime number $p$ with $p_{i} \neq p_{j}$ for $i\neq j$.

Then the compositum $E_{1}*...*E_{n}$ is finite Galois and its Galois group $\operatorname{Gal}(E_{1}*...*E_{n}/K)$ is cyclic of degree $p_{1}\dotsm p_{n}$.

The compositum is defined as $E_{1}*...*E_{n} :=(E_{1}*...*E_{n-1})*E_{n}$.

Thanks in advance for help.