1. The problem statement, all variables and given/known data Show that any field of characteristic 0 is perfect. 2. The attempt at a solution Let F be a field of characteristic 0. Let K be a finite extension of F. Let b be an element in K . I need to show that b satisfies a polynomial over F having no multiple roots. If f(x) is irreducible in F[x] then f(x) has no multiple roots. I need to show that b satisfies a irreducible polynomial in F[x]. Well, suppose b can't satisfy any irreducible polynomial in F[x]. Can I get a contradiction? What kind of element could I have that didn't satisfy any irreducible polynomial? Then how can b be in the finite extension...? A finite extension for a field of characteristic 0 is of the form F(a), it is generated by a single element. I'm stuck. I don't even know if what I've laid out so far is correct. I'm having trouble connecting the arbitrary element b to a polynomial-- It's not obvious to me that b is the root of any polynomial in F[x].