Determine whether the splitting field of a polynomial contains a subfield M such that M:$\mathbb {Q}$ is not normal For the following polynomials I need to find out if the splitting field over $\mathbb {Q}$ has a subfield M such that M:$\mathbb {Q}$ is not normal. 
1) $ x^6-7$
2)$ x^3 + 3x +3 $
3)$x^{100} - 1$
I know that by Eisenstein's criterion the first 2 are irreducible but the third is reducible. I can't think of any simple way of finding if these polynomials contain such subfields. 
I was thinking of possibly using the fundamental theorem of galois theory which states that M:$\mathbb {Q}$ is normal iff Gal(L:M) is a normal subgroup of G but I'm not sure how. 
 A: Hint: An idea here is to find if you have a complex root and a root in $R-Q$  for example $x^6-7$, you have the root $7^{1/6}$ and  $e^{2i\pi/6}$ so  $Q(7^{1/6})$ is a subfield which is not normal.
For the second, there is a real root $\alpha$ since the degree of the polynomial is 3, its derivative is $3x^2+3$ thus it is a strictly increasing real function who has only one real roots, the two others roots are complex, since $x^3+3x+3$ is the minimal polynomial of $\alpha$, $Q(\alpha)$ is not normal.
A: For the first two, you can find a real, non-rational root $\alpha \in \mathbb{R} \setminus \mathbb{Q}$. (E.g. $\alpha = \sqrt[6]{7}$ for the first one; and $\alpha$ exists for the second one by the intermediate value theorem applied to $x = -1$ and $x = 0$, and isn't rational by the rational roots theorem.) Then $Q[\alpha]:Q$ is an intermediate extension that isn't normal, for the given polynomials have $\alpha$ as roots but don't split completely over $\mathbb{Q}[\alpha]$.
For the last one, you can use the fact that the Galois group of $\mathbb{Q}[\zeta]:\mathbb{Q}$, where $\zeta$ is a primitive $100$th root of unity, is isomorphic to the group of units of $\mathbb{Z}/100$, hence abelian. Then use the fundamental theorem relating the normal extensions with the normal subgroups of the Galois group.
