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!

  • $\begingroup$ Have any thoughts? Tried? Anything you don't understand? $\endgroup$ – anon Apr 8 '14 at 1:34
  • $\begingroup$ Pretty standard result. you can find it in almost any field theory text book. $\endgroup$ – Ehsan M. Kermani Apr 8 '14 at 4:43

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