Let $K\subseteq L$ be fields and $ f \in K[X] \setminus K$. Prove that the following are equivalent.
(a) $L$ is a splitting field for $f$ over $K$
(b) $f$ splits over $L$ and $L=K(a_1,\ldots,a_n)$ where $a_1,\ldots,a_n$ are the roots of $f$ in $L$
(c) $f$ splits over some field extension of $L$ and $L = K(a_1,\ldots,a_n)$, where $a_1,\ldots,a_n$ are the roots of $f$ in that extension
I would appreciate any advice of how to prove that the statements are equivalent and how each one leads to the next!