Please provide complete rigorous proof!

Prove that K/F is a finite extension and every irreducible polynomial of F] with a root in K splits completely in K[x if and only if K is a splitting field for some polynomial p(x) E F[x] Hint: for the "if" direction: let K be a splitting field for p(x) E Flx], suppose q(x) E Fx] is irreducible over F and has a root in K, and let L be a splitting field for pq. If ?? and ?2 are roots for q(x) in L, show that [K(A) : K] K (02) K] by using the towers of fields: K (67) F(81) F(02) You will need to use the fact that splitting fields are isomorphic, which we will prove in Tuesday's class but can be found in the tertbook as Theorem 27 of Chapter 13. Prove that K/F is a finite extension and every irreducible polynomial of F] with a root in K splits completely in K[x if and only if K is a splitting field for some polynomial p(x) E F[x] Hint: for the "if" direction: let K be a splitting field for p(x) E Flx], suppose q(x) E Fx] is irreducible over F and has a root in K, and let L be a splitting field for pq. If ?? and ?2 are roots for q(x) in L, show that [K(A) : K] K (02) K] by using the towers of fields: K (67) F(81) F(02) You will need to use the fact that splitting fields are isomorphic, which we will prove in Tuesday's class but can be found in the tertbook as Theorem 27 of Chapter 13