Subgroups of a Galois group $G$

Question

I have the following exercise:

$L$ is a finite Galois extension of $K$ with Galois group $G$ (that is $G=Gal(L/K)$). Suppose $L_1$ and $L_2$ are subextensions and $G_1$ and $G_2$ are the respective subgroups of $G$. Show that $G$ is a direct product of $G_1$ and $G_2$ iff $L_1$ and $L_2$ are Galois extensions of $K$ s.t. $L_1 L_2 =L$ and $L_1 \cap L_2 =K$

What I am just wondering is the respective subgroups $G_1$ and $G_2$. If $G_1$ was Galois extension is $G_1=\operatorname{Gal}(L_1 /K)$ or is $G_1 =\operatorname{Gal}(L/L_1)$? I guessed it was the latter based on theorems for example the fundamental theorem of Galois extensions in the finite case but based on this. And is true that $G_1$ and $G_2$ are in general Galois extensions?

Answer 1

Hints (and you try to connect the dots): If $\;G\;$ is a group and $\;N_1\,,\,N_2\le G\;$ , then:

$$\begin{align*}\bullet&\;\;G=N_1\times N_2\iff\begin{cases}N_1,N_2\lhd G\\{}\\N_1N_2=G\\{}\\N_1\cap N_2=1\end{cases}\end{align*}$$

Now, using the Galois correspondence and theorems around this, and using your notation and your assumptions:

$$\begin{align*}\bullet&\;\;G=G_1G_2\iff L_1L_2=L\\{}\\\bullet&\;\;G_i\lhd G\iff L_i/K\;\;\text{is a normal extension}\iff L_i/K\;\;\text{is Galois}\\{}\\\bullet&\;\;G_1\cap G_2=1\iff L_1\cap L_2=K\;\text{(hint: what extension fits to the trivial sbgp.?)}\end{align*}$$