1. ## polynomial

For each divisor m of 6, give an example of a polynomial p(x) in Q[x], such that the splitting field E of p(x) over Q has m, ie [E:Q]=m

2. for m=3 you have to use some cubic polynomial whose determinant is a square in Q, so that the Galois group does not contain any odd permutations
for other cases, cyclotomic polynomial suffices

3. Originally Posted by FancyMouse
for m=3 you have to use some cubic polynomial whose determinant is a square in Q, so that the Galois group does not contain any odd permutations
for other cases, cyclotomic polynomial suffices
Seems like a slight typo there. It should be a "discriminant".

For a concrete example,

If $\displaystyle ch(F) \neq 2$, then the permutation $\displaystyle \sigma \in S_n$ is an element of $\displaystyle A_n$ if and only if it fixes the square root of the discriminant D.

We know that A_3 = Z_3. So the possible polynomial can be

$\displaystyle f(x) = x^3 + x^2 - 2x -1 \in \mathbb{Q}[x]$ and its discriminant is 49. Then, if the splitting field of f(x) is K, [K:Q]=3 and the order of Gal(K/Q) is 3.

4. >"discriminant".
Ah yes, thanks for pointing out.