The following Daily Questions were asked this week.

## Monday, May 8

## First Question

**True** or **False**: If $\phi:F\to D$ is a finite field extension and $H\leq \operatorname{Aut}_F(D)$, then $D^H\subseteq D$ is a Galois extension.

## Reveal Answer

### Answer

True.

## Second Question

Given a Galois extension $\phi:F\to D$ and an element $\alpha\in D$, the **Galois conjugates** of $\alpha$ are…

## Reveal Answer

### Answer

…the elements $\sigma(\alpha)$ for $\sigma\in\operatorname{Gal}_F(D)$.

## Tuesday, May 9

For today’s questions, assume $F\subseteq D$ is a Galois extension and $F\subseteq E\subseteq D$ is an intermediate extension.

## First Question

The extension $F\subseteq E$ is Galois if and only if…

## Reveal Answer

### Answer

…$\operatorname{Aut}_E(D)$ is a *normal* subgroup of $\operatorname{Aut}_F(D)=\operatorname{Gal}_F(D)$.

## Second Question

If $F\subseteq E$ is Galois, then $\operatorname{Gal}_F(E)$ is isomorphic to…

## Reveal Answer

### Answer

…the quotient group $\operatorname{Gal}_F(D)/\operatorname{Gal}_E(D)$.

## Thursday, May 11

For today’s questions, suppose $F\subseteq D$ is a finite Galois extension and $F\subseteq E\subseteq D$ is an intermediate extension.

## First Question

The set of cosets $\operatorname{Gal}_F(E)/\operatorname{Gal}_E(D)$ is in natural bijection with…

## Reveal Answer

### Answer

…$\operatorname{Mor}_F(E,\overline{F})$, the set of all field morphisms from $E$ over $F$. (Such maps are all isomorphisms of $E$ onto their image.)

## Second Question

The extension $F\subseteq E$ is Galois if and only if every field isomorphism from $E$ over $F$ is…

## Reveal Answer

### Answer

…an automorphism of $E$.

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