Question

Question 1 (15 points) (a) Determine whether the following extensions are normal. Justify your answer. (1) Q CQ(V2, 92). (2) F = F3(t) C F(a), where t is a variable and a is a root of x3 – t in the splitting field. (b) Give an example of a normal extension of Q that is not finite. Justify your answer.

Answer 1

(c) let fu... fre KEXJ be polynomials in the variable ona the field K. Then the splitting field N of fe..., for in obtained by adjoining all of the roots of fi to K. In particular Nlk is a finitely generated, algebric extension. and thus finite. Moreova Nis splitting field of one polynomial, namely fo... fr. So the normal extension N/K. you are looking for must be of infinite degree. You can take the algebraic closure a of the nationals o for example. It is normal by definition and it is infinite because the polynomial X-2 are irreducible for every nEN. K= Q(J2, 53) is a normal extension over a. A basic theorem in Galois theasy say that a finite extension Epf is normal and seperable if and only if E is the splitting field of a seperable polynomial f(x) E F[x] It suffices to consider, f(x) = (x²-2)(x²-3). Its splitting field must contain 52 and 53 and since K is the smallest field containing both of them, I must in fact be the splitting field of f(x) Therefore K is normal over a