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.
$\endgroup$ 51 Answer
$\begingroup$Note the following segment from the fundamental theorem of galois theory
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.
$\endgroup$
