The following Daily Questions were asked this week.

Tuesday, April 25

First Question

True or False: The cyclotomic polynomial $\Phi_n(x)$ is irreducible in ${\bf Q}[x]$.

Reveal Answer

Answer

True.

Second Question

True or False: All of the nonzero coefficients in the polynomial $\Phi_n(x)$ are $\pm 1$.

Reveal Answer

Answer

False. Although this might appear to be true for the first many values of $n$, it is not true for all $n$. The first instance of a nonzero coefficient other than $\pm 1$ is at $n=105$, in which the polynomial $\Phi_{105}(x)$ has a coefficient of -2.

Thursday, April 27

For today’s questions, suppose $F\subseteq D$ is a field extension.

First Question

If $\alpha\in D$ is algebraic over $F$, then for every $\sigma \in \operatorname{Aut}_F(D)$ the element $\sigma(\alpha)$ is…

Reveal Answer

Answer

…a root in $D$ of $m_{\alpha, F}(x)$.

Second Question

If $H\leq \operatorname{Aut}_F(D)$, then $D^H$ denotes…

Reveal Answer

Answer

…the fixed field of $H$ in $D$, i.e., the subfield of $D$ consisting of all elements fixed by every $\sigma\in H$. In set notation,
\[
D^H = \{\alpha\in D\mid \sigma(\alpha)=\alpha \text{ for all }\sigma \in H\}.
\]