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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 $r^{-1} (c)$.2012-12-15
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    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 $[K:\mathbb Q(\omega^3)]$?2012-12-15
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    @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 $a\cdot r(x)$.2012-12-15
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    But how do you guarantee $q (x)$ has rational coefficients?2012-12-16
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    @HenningMakholm I know that $[Q(w):Q] = 12$ and that $[Q(w^3):Q] = 4$ since $w^3$ generates a subgroup of order 4 of the multiplicative group of 12th roots of unity and also because it satisfies the equation $x^4-1=0$.2012-12-16
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    @ZhenLin ahh yes. thank you for pointing that out. So how would you go about solving this question then? can you give me a hint?2012-12-16
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    @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?2012-12-16
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    @AdenDong That's right. Edit: oh, and I see you answered your second question already...2012-12-16

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