I wanted to check my solutions for this problem:
Let $f ∈ K[T]$ with prime factorization $f = a·p_1^{m_1}\cdots p_t^{m_t}$ , with $a ∈ K^{×}$ and pairwise different normalized and irreducible polynomials $p_i$.
To show: The splitting field of $f$ is identical to that of the polynomial $p_1\cdots p_t$
In my opinion the answer is really basic which is why I have doubts:
We can ignore the constant $c$ since $c \in K^{x}$. If we take the polynomials individually, being irreducible and different from each other, each polymial $p_i$ has the roots $a_1, \dots , a_{ji}$. These roots do not change if the polynomial is raised to a power. It therefore goes without saying that the splitting field of $f$ is identical to that of the polynomial $p_1\cdots p_t$