Let be a field with characteristic 0. is a finite field extension of . Prove that is a separable extension...
I know that for an , if the minimal polynomial splits on , then is separable on . But why splits?
Let be a field with characteristic 0. is a finite field extension of . Prove that is a separable extension...
I know that for an , if the minimal polynomial splits on , then is separable on . But why splits?
You are misunderstanding what "seperable" means. An irreducible polynomial is "seperable" over iff has no repeated roots in its splitting field. Now is seperable iff the minimal polynomial for is seperable over . This does not mean that the miniminal polynomial must split over .