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Here is my recent homework question:

For each of the following five fields $F$ and five groups $G$, find an irreducible polynomial in $F[x]$ whose Galois group is isomorphic to $G$. If no example exists, you must justify that.

Fields $F$: $\mathbb{C}$, $\mathbb{R}$, $\mathbb{F}_{11}$, $\mathbb{Q}$, $\mathbb{Q}(i)$

Groups $G$: $C_2$, $C_5$, $C_2\times C_2$, $S_3$, $D_4$

I've found the polynomials for first 4 fields; however, I've got no idea about the $\mathbb{Q}(i)$ one.

Can anyone here help me? Thanks, and regards.

Now, I just found I made mistakes in looking for C2xC2 and C5 one . The polynomial I found are not irreducible ( in fact only with separable irreducible factors ). Moreover, I don't know how to check the irreducibility of polynomial in F11. So I also can't do this part..

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    Seems to me that if you could answer the question for $\mathbb{Q}$, it should be pretty much the same for ${\mathbb{Q}}(i)$. You are being asked so supply 25 polynomials (or argue that none exists in the particular case), right? Did you have any trouble with $\mathbb{Q}$ and $C_5$?2012-05-29
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    No, the subtlety is that $\mathbb{Q}(i)$ has very different irreducible polynomials than $\mathbb{Q}.$ In particular, you're going to be looking for polynomials with real roots, or complex roots of the form $a+bi$ with either $a$ or $b$ in $\mathbb{R}$.2012-05-30
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    @rotskoff, well, admitted; but the polynomial I found for $\mathbb{Q}$ and $C_5$ works equally well for ${\mathbb{Q}}(i)$, because it’s Eisenstein for the prime $11$.2012-05-30
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    @Lubin : I found I made mistakes in looking for polynomial in Q[x] whose galois group isormorphic to C2xc2 , C5 . For isomorphic to C2xC2, the only polynomial I can think of is of(X^2 - D1)(x^2 -D2) form , where D1 , D2 are not square in Q . However , this should be reducible in Q .. For C5 , the polynomial I found is reducible too.. Some of my classmates think there is no such irreducible polynomial over Q in C2xC2 and C5 case .. Is that correct ? I am wondering if there is some systematic way to find these polynomials .2012-05-30
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    Maybe the inverse direction could be helpful?Per chance you could consult this:http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/cubicquartic.pdf2012-05-30

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