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: $[D:D^H] = |H|$.