# A few questions on field theory

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?

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I suspect $r (c)^{-1}$, i.e. $1 / r(c)$, is meant instead of $r^{-1} (c)$. – Zhen Lin Dec 15 '12 at 23:48
If $K$ is a field strictly between $\mathbb Q(\omega)$ and $\mathbb Q(\omega^3)$, then what are the possible pairs of values of $[\mathbb Q(\omega):K]$ and $[K:\mathbb Q(\omega^3)]$? – Henning Makholm Dec 15 '12 at 23:53
@ZhenLin if that's the case, wouldn't the solution be trivial? just take r(x) to be any non-zero polynomial and q(x) to be $a\cdot r(x)$. – Aden Dong Dec 15 '12 at 23:55
But how do you guarantee $q (x)$ has rational coefficients? – Zhen Lin Dec 16 '12 at 0:00
@HenningMakholm So continuing from above, $[Q(w):Q(w^3)] = 3$. So the only possibilities for $[Q(w):K]$ and $[K:Q(w^3)]$ are 1, 3 or 3, 1. So $K$ must be either $Q(w)$ or $Q(w^3)$. So in other words there cannot be a field strictly between $Q(w)$ and $Q(w^3)$. Is this line of thought correct? – Aden Dong Dec 16 '12 at 0:10