I am preparing for my final exam for Abstract Algebra using the book written by Michael Artin. I have some questions.
Q1) Find the splitting field of $x^4 + 1$ over $\mathbb{Q}$. So I found a solution where it says the roots are $e^{i \pi/4}, e^{3i \pi/4}, e^{5i \pi/4}, e^{7i \pi/4}$. Could some explain to me how they obtain this using the roots of unity formula?
I used the alternative way by decomposing $x^4 + 1 = (x^2 + i)(x^2 - i)$ and got $x = \pm i \sqrt{i}$ and $\pm \sqrt{i}$. But don't these roots make sense? For the record, using these roots will get me [$\mathbb{Q}$(these roots):$\mathbb{Q}$] = 2. (Different than the one above where the dimension = 4)
Q(2): Determine all automorphisms of the field $\mathbb{Q}(2^ {1/3})$ and of the field $\mathbb{Q}(2^{1/3}, \epsilon)$, where $\epsilon = e^ {2i \pi/3}$
Could you someone give me hints on how to find these automorphisms?
Thank you for your help!