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

Exercise 5 (Minimal polynomials 1; 15 points): Acomplex num- ber 05 is called algebraic if there is a nonzero monic polynomial with rational coefcients m0, so that molar) = 0. Such a poly- nomial of smallest degree is called the minimal polynomial of oz. As an example, x/i is algebraic since it is a root of 3:2 2, which is its minimal polynomial; whereas 7r is not algebraic. (1) Show that mo, is irreducible and that mo, divides any poly nomial f E QM that has oz as a root. (2) Show that a: is algebraic if and only if the Qlinear span of {1, 0:, 0:2, . . .} is nitedimensional. Call this span (2(a). (Hint: Write a proof by induction to show that if d 2 deg ma, then oak is in the span of {1, oz, . . . ,Odd_1} for all 1%. Note that mo, gives a way of expressing oz"! as an element of this span.) (3) Show that if oz is algebraic, then Q(a) is a eld. (Remem ber: since Q(Ol) is contained in (C, which we know is a eld, you just have to check that (2(a) is closed under addition, multiplication, and taking additive and multiplicative in- verses. The hardest part will be showing closure under multiplicative inverses. Since every element of Q(a) can be written as f(a) for some f E Q x], I would first recom- mend showing that if f(a) * 0, then (f, ma) = Q[x])