1) If $P\in K[X]$ irreducible, and if $\alpha_1,...,\alpha_n$ are his roots, does his splitting field is $K(\alpha_1,...,\alpha_n)$ ?
2) If $P(X)=P_1(X)\cdot ...\cdot P_n(X)\in K[X]$ where $P_i$ are irreducible and the roots of $P_i(X)$ are $\alpha_1^i,...,\alpha_{m_i}^i$, does $$K(\alpha_1^1,...,\alpha_{m_1}^1,\alpha_1^2,...,\alpha_{m_2}^2,...,\alpha_1^n,...,\alpha_{m_n}^n)$$ is the splitting field of $P(X)$ ?
In both case, I think it's true, but I would like to have a confirmation.