# separable extension and separable polynomial

• Aug 30th 2009, 07:18 AM
ynj
separable extension and separable polynomial
If $E$is a splitting field over $F$of a separable polynomial, why $E$is a separable extension of $F$ which means $\forall x\in E$, $x$is a root of a separable polynomial over $F$?
• Aug 30th 2009, 06:15 PM
NonCommAlg
Quote:

Originally Posted by ynj
If $E$is a splitting field over $F$of a separable polynomial, why $E$is a separable extension of $F$ which means $\forall x\in E$, $x$ is a root of a separable polynomial over $F$?

the proof of this theorem can be found in any textbook in field theory. the idea is to prove, by induction over $[E:F],$ that $|\text{Aut}_F(E)|=[E:F].$
• Aug 30th 2009, 08:22 PM
ynj
Em..I am sorry that I do not fully understand whay my book is saying..but now I understand...