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.

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    $\begingroup$ You don't need normality of $H$ unless you want $\Bbb M$ to be Galois over $\Bbb K$. That's not something you're asked for. $\endgroup$ – Arthur Oct 30 '18 at 17:45
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    $\begingroup$ The title is not good. Subfields of groups? $\endgroup$ – Dietrich Burde Oct 30 '18 at 19:07
  • $\begingroup$ Sorry, I confuse the order of "Groups "and "extension", I edited. $\endgroup$ – Eduardo Silva Oct 30 '18 at 19:54
  • $\begingroup$ @Arthur, so, I can conclude the exercise with my idea above and the use it to prove the aplication i mentioned? $\endgroup$ – Eduardo Silva Oct 30 '18 at 19:57

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