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 $E$.