This question, the one on the primitive element and perhaps others are very basic, standard stuff...I mean, are you trying to cope with this material without the aid of books? Any decent book in algebra/fields/extensions/Galois Theory (with introduction on extensions) deal with this (for example, you could first prove that a field K of characteristic p is perfect iff

the p-th power map is an automorphism of K.)

I propose you grab some books, try to get some help from them AND THEN, if you get stuck somewhere, you ask.

I firmly believe that anyone studying mathematics at any level above high school (and also before is highly recommended) must get used to books and must have several by his/her side all the time, either of his property or the library's.

Tonio