Conditions required for Galois Correspondence This question is based on Theorem 10.2 of John Howie's 'Fields and Galois Theory', p.171. All fields have characteristic zero.
$M$ is a normal radical field extension of $K$, i.e. $M=K(\alpha_1,\alpha_2,...,\alpha_n)$ where $\alpha_i^{p_i}\in K(\alpha_1,...\alpha_{i-1})$ for some prime $p_i$.
We also define $P=M(\omega)$ where $\omega$ is a primitive $p_i$th root of unity. Howie points out that, as $P$ is a splitting field for $X^{p_i}-1$ over $M$, $P/M$ is a normal extension of $M$. So far so good.
As we have $M/K$ normal then, using the Fundamental Theorem of Galois Theory (FTGT), we have $\text{Gal}(P/M)\vartriangleleft \text{Gal}(P/K)$. Also by FTGT we have the following isomorphism:
$$\text{Gal}(M/K)\simeq\frac{\text{Gal}(P/K)}{\text{Gal}(P/M)}$$
However, in order for the FTGT to apply, I understand that we also require $P$ to be a normal extension of $K$. Howie does not seem to be explicit about why this condition holds. The only strategy I know would be to show that $P=K(\omega,\alpha_1,\alpha_2,...,\alpha_n)$ is a splitting field for a polynomial over $K$, however it is not obvious to me what that polynomial would be, or how I could show that its splitting fields was the one required.

The vertical composition of normal field extensions isn’t normal, but the horizontal compositon is.

Let $E_1/F$ end $E_2/F$ be finite field extensions of a common base field $F$. If $E_1/F$ and $E_2/F$ are normal, then $E_1E_2/F$ is normal.

Proof. Let $f_1, f_2 ∈ F[X]$ be polynomials such that $E_1$ and $E_2$ are splitting fields of $f_1$ and $f_2$ respectively. Then $E_1E_2$ is a splitting field of $f_1f_2 ∈ F[X]$.
Here, $M(ω)/K$ is a horizontal composition of normal field extensions, by $M(ω) = K(ω)M$.
A counterexample for the failure of the vertical composition of normal field extensions to be normal is $ℚ(\sqrt[4] 2)/ℚ$ with its intermediate field $ℚ(\sqrt 2)$.
