The following Daily Questions were asked this week.

## Monday, May 1

## First Question

We say a finite field extension $\phi:F\to E$ is **Galois** if…

## Reveal Answer

### Answer

…$|\operatorname{Aut}_F(E)| = [E:F]$. (We will soon see this is equivalent to $\phi:F\to E$ being a splitting field extension of a separable polynomial $f\in F[x]$.)

## Second Question

The **Galois group** of a Galois field extension $\phi:F\to E$ is…

## Reveal Answer

### Answer

…the group $\operatorname{Aut}_F(E)$, usually denoted $\operatorname{Gal}_F(E)$ (or $\operatorname{Gal}(E/F)$).

## Tuesday, May 2

## First Question

**True **or **False**: If $\phi:F\to E$ is a splitting field extension of a separable polynomial $f(x)\in F[x]$, then $\phi$ is a Galois extension.

## Reveal Answer

### Answer

True. (Soon we’ll see the converse is also true.)

## Second Question

**True** or **False**: Every finite extension $\phi:{\bf F}_p\to F$ is a Galois extension, with cyclic Galois group.

## Reveal Answer

### Answer

True. Moreover, the cyclic Galois group is generated by the Frobenius automorphism.

## Thursday, May 4

## First Question

For a group $G$ and field $L$, a **character of $G$ with values in $L$** is…

## Reveal Answer

### Answer

…a group (homo)morphism $\sigma:G\to L^{\times}$.

## Second Question

Suppose $\phi:F\to D$ is a finite field extension and $H\leq \operatorname{Aut}_F(D)$. How are the numbers $[D:D^H]$ and $|H|$ related?

## Reveal Answer

### Answer

They are equal: $[D:D^H] = |H|$.

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