Assignment:
Show that any algebraic extension of a perfect field is perfect (using the below hint only).
Hint: Let K be a perfect field and F an algebraic extension of K. If F is not perfect, then there is a polynomial f(x) an element of F[x] that has an irreducible factor p(x) with a repeated root u. Here u is algebraic over K; let g(x) be the minimal polynomial of u over K. What is the relationship between g(x) and p(x)? Show that g(x) has a repeated root - this contradicts the hypothesis that K is perfect.
Provide complete and step by step solution for the question and show calculations and use formulas.