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..