# separable extension and separable polynomial

• Aug 30th 2009, 06:18 AM
ynj
separable extension and separable polynomial
If $\displaystyle E$is a splitting field over $\displaystyle F$of a separable polynomial, why $\displaystyle E$is a separable extension of $\displaystyle F$ which means $\displaystyle \forall x\in E$,$\displaystyle x$is a root of a separable polynomial over $\displaystyle F$?
• Aug 30th 2009, 05:15 PM
NonCommAlg
Quote:

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

the proof of this theorem can be found in any textbook in field theory. the idea is to prove, by induction over $\displaystyle [E:F],$ that $\displaystyle |\text{Aut}_F(E)|=[E:F].$
• Aug 30th 2009, 07:22 PM
ynj
Em..I am sorry that I do not fully understand whay my book is saying..but now I understand...