The following Daily Questions were asked this week.

First Question

True or False: If $F\subseteq E$ is a Galois field extension, $F\subseteq F’$ is finite field extension, and $E’=EF’$ is the composite field, then the field extension $F’\subseteq E’$ is Galois.

True. Moreover, $\operatorname{Gal}_{F’}(E’)\cong \operatorname{Gal}_{E\cap F’}(E)$.

Second Question

Suppose $F\subseteq E_1$ and $F\subseteq E_2$ are both Galois field extensions. Which of the following field extensions below are Galois?

1. $F\subseteq E_1E_2$
2. $F\subseteq E_1\cap E_2$
3. $E_1\subseteq E_1E_2$
4. $E_2\subseteq E_1E_2$
5. $E_1\cap E_2 \subseteq E_1E2$

All of them.

First Question

True or False: Every finite extension is simple.

In terms of intermediate extensions, a finite extension $F\subseteq E$ is simple if and only if…
…there are only finitely many distinct intermediate extensions $F\subseteq K\subseteq D$.