The following Daily Questions were asked this week.

## Tuesday, May 16

## 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.

## Reveal Answer

### Answer

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?

- $F\subseteq E_1E_2$
- $F\subseteq E_1\cap E_2$
- $E_1\subseteq E_1E_2$
- $E_2\subseteq E_1E_2$
- $E_1\cap E_2 \subseteq E_1E2$

## Reveal Answer

### Answer

All of them.

## Thursday, May 18

## First Question

**True** or **False**: Every finite extension is simple.

## Reveal Answer

### Answer

False. Every finite *separable* extension is simple.

## Second Question

In terms of intermediate extensions, a finite extension $F\subseteq E$ is simple if and only if…

## Reveal Answer

### Answer

…there are only finitely many distinct intermediate extensions $F\subseteq K\subseteq D$.

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