The following Daily Questions were asked this week.

Monday, May 22

First Question

True or False: Every finite abelian group is (isomorphic to) the Galois group of a Galois extension ${\bf Q}\subseteq E$.

Reveal Answer

Answer

True.

Second Question

Given a finite abelian group $G$, we constructed a Galois extension ${\bf Q}\subseteq E$ with Galois group isomorphic to $G$ as an intermediate extension of what type of field extension of ${\bf Q}$?

Reveal Answer

Answer

cyclotomic extension, i.e., an extension of the form ${\bf Q}\subseteq {\bf Q}(\zeta_n)$.

Tuesday, May 23

First Question

True or False: In our construction, we showed that for each field $F$ there is a Galois extension $F\subseteq E$ with Galois group isomorphic to $S_n$.

Reveal Answer

Answer

False. We actually showed that the field extension $F(s_1,\ldots , s_n)\subseteq F(x_1,\ldots, x_n)$ is Galois with Galois group isomorphic to $S_n$.

Second Question

True or False: For every finite group $G$, there is a Galois extension $F\subseteq E$ with Galois group isomorphic to $G$.

Reveal Answer

Answer

True.

Thursday, May 25

First Question

A polynomial $f(x)\in F[x]$ is separable if and only if its discriminant $D_f$ is …

Reveal Answer

Answer

… nonzero.

Second Question

Suppose $F\subseteq E$ is a splitting field extension of a separable polynomial $f(x)\in F[x]$. The Galois group of this extension is isomorphic to a subgroup of $A_n$ if and only if …

Reveal Answer

Answer

… $\Delta_f\in F$, i.e., $D_f$ is the square of an element in $F$.