This problem was asked in my Abstract algebra assignment and I was unable to solve it so I am asking for help here.
Let E be an intermediate field .
(a) If u $\in$ F is separable over K, then u is separable over E.
(b) If F is separable over K, then F is separable over E and E is separable over K.
I tried (a) but couldn't solve it. The irreducible polynomial of u say f is given seperable over K which means that it only has single root. But I have no reason to believe why it is also seperable over E.
For (b) F is seperable over K means that all elements of F are seperable over K then using (a) I can deduce that F is seperable over E but why E must be seperable over K. Also if x$\in E$ implies that $x \in F$ and so x is separable over K. So it's done . But for (a) i need help.
Kindly help me with this question!