Let L be the splitting field in C for x3 2 Qx Prove that L

Let L be the splitting field in C for x^3 - 2 Q[x]. Prove that [L: Q] = 6. (We covered this in class, but 1 want you to write up a careful proof with details included. You do not have to compute Aut(L).)

Solution

Let c denote the real cube root of 2. (c is unique , as there is only one real root of the polynomial X³-2, as can be seen either by graphing, or using the discriminant)

Then the other roots of the polynomial are the roots of a quadratic equation over Q(c).

So the field L has degree 2 over Q(c)

As Q(c) has degree 3 over Q, it follows that L has degree 6 over Q