Let $\mathbb{K}\subseteq \mathbb{L}$ be a Galois extension with order $n$. If $p$ is a prime divisor of $n$, show that exists a subfield $\mathbb{M}$ of $\mathbb{L}$ such that $[\mathbb{L},\mathbb{M}]=p$.
My try:
Since $\mathbb{L}$ is Galois, $o(Gal(\mathbb{K},\mathbb{L}))=n$ and by Cauchy Theorem, there is a group $H \subseteq Gal(\mathbb{K},\mathbb{L})$ with such order.
If I prove that $H$ is a normal subgroup of $Gal(\mathbb{K},\mathbb{L})$, then the Theorem of Galois correspondence give me subfield of $\mathbb{L}$ with the desired order, and by the tower law i can prove that if $n=pq$, then $[\mathbb{M},\mathbb{K}]=q$.
My doubt is how to ensure the normality of $H$ to conclude my proof?
An aplication of this exercise is to the the unique existence subfields $\mathbb{K}_{1}, \mathbb{K}_{2}$ of $\mathbb{Q}(\zeta_7)$ in $\mathbb{Q}$ such that $[\mathbb{\mathbb{K}_{1}},\mathbb{Q}]=2$ and $[\mathbb{\mathbb{K}_{1}},\mathbb{Q}]=3$.
I would like to understand the first problem to solve and understand the second.