DO NOT USE AI TO ANSWER THE QUESTION! I WILL REPORT YOU IF I FIND OUT.

Exercise 6 (Minimal polynomials 2; 15 points): Suppose that or E (C is algebraic. (1) Let Mat be the vector space of linear maps from Q(oz) to itself. Let 1/; : Q(a) > Mat be the map that sends a number H to the linear map T5 : Q(oz) > Q(oz) Where T5(y) = ,8 . 3;. Show that w is an injection and that if f E QM, then f (T3) = T)' (Hint: It sufces to show that 1131-1-32 : T31 + T52 and that 219152 : T512732\") (2) Show that if ,8 E Q(oe), then the minimal polynomial of the number ,6 is the same as the minimal polynomial of the matrix T3. (3) Show that if f and g are irreducible polynomials in QM and f 75 9, then no root of f is a root of g. (4) Show that if 6 Q(a) then the characteristic polynomial of the matrix T5 is a power of the minimal polynomial of the number 5. (Hint: use the previous two parts, unique prime factorization of polynomials, and the fact that, if A is a matrix, then any root of X A is a root of m A.)