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
For a concrete example,
If , then the permutation is an element of 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
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.