# Subfields of Galois Extensions and association with Galois Groups

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.

• 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. – Arthur Oct 30 '18 at 17:45
• The title is not good. Subfields of groups? – Dietrich Burde Oct 30 '18 at 19:07
• Sorry, I confuse the order of "Groups "and "extension", I edited. – Eduardo Silva Oct 30 '18 at 19:54
• @Arthur, so, I can conclude the exercise with my idea above and the use it to prove the aplication i mentioned? – Eduardo Silva Oct 30 '18 at 19:57