#### This Is A Custom Widget

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#### This Is A Custom Widget

This Sliding Bar can be switched on or off in theme options, and can take any widget you throw at it or even fill it with your custom HTML Code. Its perfect for grabbing the attention of your viewers. Choose between 1, 2, 3 or 4 columns, set the background color, widget divider color, activate transparency, a top border or fully disable it on desktop and mobile.

# The Daily Questions of Week 9

///The Daily Questions of Week 9

## The Daily Questions of Week 9

The following Daily Questions were asked this week.

## First Question

Suppose $F\subseteq E$ is the splitting field extension of a separable polynomial $f(x)$ of degree $n$. If $f(x)$ is irreducible in $F[x]$, then $\operatorname{Gal}_F(E)$ is (isomorphic to) what type of subgroup of $S_n$?

transitive subgroup. (The converse statement is also true.)

## Second Question

Suppose $f(x)\in {\bf Q}[x]$ is a monic irreducible quartic polynomial. Let ${\bf Q}\subseteq E$ be a splitting field extension of $f(x)$, and let $h(y)$ be the resolvent cubic of $f(x)$. If $h(y)$ splits completely in ${\bf Q}[y]$, then $\operatorname{Gal}_{\bf Q}(E)$ is isomorphic to …

… the Klein 4-group, i.e., $({\bf Z}/2{\bf Z})\times ({\bf Z}/2{\bf Z})$.

## Friday, June 2

For today’s questions, suppose $F$ is a field of characteristic not dividing $n$ that contains the $n$th roots of unity.

## First Question

For every $a\in F$, the extension $F\subseteq F(\sqrt[n]{a})$ is a Galois extension with what type of Galois group?

A cyclic Galois group, of ordering dividing $n$.

## Second Question

If $F\subseteq E$ is a cyclic extension of degree $n$, then what can we say about $E$?

It is a simple extension of the form $E=F(\sqrt[n]{a})$ for some $a\in F$. (We can even use Lagrange resolvents to help us find $\sqrt[n]{a}$.)