Question 1: Let w be a primitive 12th root of unity. How many subfields are there strictly between $Q(w^3)$ and $Q(w)$? I have no idea how to do this and don't even know where to start. A hint would be much appreciated.
Question 2: Find a primitive element of the splitting field of x^4-2 over Q. The splitting field can easily be found to be $Q(\sqrt[4]{2}, i)$. I also know that since Q has characteristic zero, by Steinitz this splitting field should be a simple extension of Q. However I'm not sure how to explicit find this primitive element.
edit: I just went through the proof of Steinitz and found how to explicitly find this primitive element.
Question 3: Let a and b be the positive roots of polynomials $(x+1)^2-2$ and $(x+2)^2-5$. Let $c=a+b$. Part one says to find a polynomial $p(x) \in Q[x]$ such that $p(c) = 0$. This is straightforward. Part two says to find a couple of polynomials $q(x), r(x) \in Q[x]$ such that $q(c)r^{-1}(c) = a$. I don't even get what the question is asking for. What exactly is $r^{-1}(c)$? Also, how would you go about finding a pair of such polynomials?