# The fundamental theorem of Galois theory

Let $E/Q$ be a Galois extension of degree $p^2$, where $p$ is a prime number. Prove that $L/Q$ is a Galois extension for any $L \in Intermediate(E/Q)$ and find $p$ if the cardinality of $Intermediate(E/Q)$ is 100.

I'm trying to apply The fundamental theorem of Galois theory but not sure where to start.

• What's the group-theoretic condition on an intermediate extension $L/Q$ of $E/Q$ that ensures $L/Q$ is Galois (group-theoretic in reference to the corresponding subgroup $\mathrm{Gal}(L/E)\subseteq\mathrm{Gal}(L/Q)$)? Commented Feb 26, 2015 at 14:20
• @KeenanKidwell So you mean finite p-subgroup? Since the degree is $p^2$, we can have a chain of groups then use the fundamental theorem of Galois theory? Commented Feb 26, 2015 at 14:29
• Your Galois group $\mathrm{Gal}(L/Q)$ has order $p^2$. A proper, non-trivial subgroup of such a group is of order $p$. What can you say about subgroups of index $p$ in a group of order $p^2$? Commented Feb 26, 2015 at 14:58
• Recall that every group of order $p^2$ is abelian, and all subgroups of abelian groups are normal. Commented Feb 26, 2015 at 18:01
• Does the set Intermediate$(E/\Bbb{Q})$ contain both $E$ and $\Bbb{Q}$ as well? Because the word intermediate appears there I would suspect that it does not. But, unless it does, there is no such $p$? Commented Mar 1, 2015 at 20:24

Let $L$ be an intermediate field of $E/Q$, then $L/Q$ is galois $\iff Gal(E/L)\trianglelefteq Gal(E/Q)$
That is, $L/Q$ is a galois extension when the galois group for $E/L$ is a normal subgroup of the galois group for $E/Q$. Since $E/Q$ is galois, $|Gal(E/Q)|=[E:Q]=p^2$. Any group with order $p^2$, where $p$ is prime, is abelian(a proof is provided here) and as $Gal(E/Q)$ is abelian, all of its subgroups, must be normal subgroups(a proof for this is provided here if you're interested).
As $Q\subseteq L\subseteq E$, $Gal(E/Q)\subseteq Gal(E/Q)$, therefore $Gal(E/L)$ must be a normal subgroup of $Gal(E/Q)$, and by segment from the fundamental theorem of Galois theory, $L/Q$ is a galois extension.