My understanding is that every countable Abelian $p$-group can be written as a direct sum of at most countably many finite cyclic groups and at most countably many copies of the Prüfer $p$-group. Suppose that when such a direct-sum decomposition is written for a particular countable Abelian $p$-g...

Let $E_1/F$ and $E_2/F$ be two radical extensions. Then $E_1E_2/F$ is also radical extension. Since $E_1$, $E_2$ are radical extensions over $F$, there is $F\subset F(\alpha_1)\subset\cdots F(\alpha_1,...,\alpha_n) =E_1$ such that $\alpha_1^{n_1}\in F$ and $\alpha_i^{n_i}\in F(\alpha_1,...,\alp...