\documentclass[12pt]{amsart}
\title{Algebra 2. Homework 11}
\author{}

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\begin{document}
\maketitle

\problem{1} Consider the following subgroups of $\C^\times$.
\[
\mu_n(\C) := \{z\in\C : z^n=1\},\qquad
\mu_\infty(\C) := \{z\in\C : z^r=1\text{ for some } r\in\Z_{\geq 1}\}.
\]

Let $\mathrm{Z}=\Z_{\geq 1}$ with partial order given by divisibility.

\begin{enumerate}
\item Show that $\{\mu_n(\C)\}_{n\in\mathrm{Z}}$ is a direct system, and
$\mu_\infty(\C) = \Dlim{n\in\mathrm{Z}}\mu_n(\C)$.\\

\item Show that $\mu_\infty(\C) \cong \Q/\Z$.\\

\item Show that $\Q(\mu_\infty) = \Dlim{n\in\mathrm{Z}} \Q(\mu_n)$.\\

\item Using $\Gal{\Q(\mu_n)}{\Q}\cong \lp\Z/n\Z\rp^\times$, show that
$\Gal{\Q(\mu_\infty)}{\Q}\cong \Ilim{n\in\mathrm{Z}} \lp\Z/n\Z\rp^\times$.
Describe a fundamental system of open neighbourhoods of $\Id\in\Gal{\Q(\mu_\infty)}
{\Q}$.\\
\end{enumerate}

\problem{2} Recall the definition of cyclotomic polynomials.
Let $P_n\subset \ol{\Q}$ be the set of primitive $n^{\sth}$ roots of unity.
Then
$\{\Phi_n(x)\}_{n\geq 1}$ is defined as:
\[
\Phi_n(x) = \prod_{a\in P_n} x-a
\]
Note that $\Phi_1(x)=x-1$.\\

\begin{enumerate}
\item Show that $\ds x^n-1 = \prod_{d|n} \Phi_d(x)$. Use this to show
that $\Phi_d(x)\in\Z[x]$.\\

\item Let $p\geq 2$ be a prime number. Compute $\Phi_p(x)$. Show
that $\Phi_{p^r}(x) = \Phi_p(x^{p^r})$.\\

\item Let $\mu : \Z_{\geq 1} \to \{0,\pm 1\}$ be defined by:
\[
\mu(p_1^{k_1}\ldots p_h^{k_h}) = \left\{
\begin{array}{ll}
0 & \text{if } k_j\geq 2 \text{ for some } 1\leq j\leq h \\
(-1)^h & \text{otherwise}
\end{array}
\right.
\]
Prove that $\ds \Phi_n(x) = \prod_{d|n} \lp x^{\frac{n}{d}} - 1\rp^{\mu(d)}$.\\

\item Compute $\Phi_{10}(x)$ and $\Phi_{12}(x)$.\\
\end{enumerate}

\problem{3} Let $L/K$ be a finite Galois extension. For $\alpha\in L$,
prove that
$L=K(\alpha)$ if and only if $\mathrm{Stab}_{\Gal{L}{K}}(\alpha)=\{\Id\}$.\\

\problem{4} Let $K$ be a field of characteristic $p$, such that
$K$ contains a primitive $n^{\sth}$ root of unity. Prove that $p$ does not
divide $n$.\\

\problem{5} Show that $x^p-x-a\in\FF_p[x]$ is irreducible for
every $a\in\FF_p^\times$.\\

\problem{6} Give an example of a finite field extension $L/K$
such that there are infinitely many intermediate extensions $K\subset E\subset L$.\\

\problem{7} Let $K=\FF_p(\lambda)$ and $L=\FF_p(\mu)$ where $\mu^p=\lambda$.
Show that, for $\alpha\in L$:
\[
\mathsf{N}_{L/K}(\alpha) = \alpha^p,\qquad
\operatorname{Tr}_{L/K}(\alpha)= 0.
\]
\ \\

\problem{8} Let $E/K$ be a finite extension. Show that $E/K$
is separable if and only if $E\times E \to K$, $(x,y)\mapsto \operatorname{Tr}_{E/K}(xy)$
is a non--degenerate, symmetric $K$--bilinear form.
Show that the above condition is equivalent to the existence
of $a\in E$ such that $\operatorname{Tr}_{E/K}(a)\neq 0$.\\

\begin{center}
{\em In Problems 9 and 10 below, $m\in\Z_{\geq 1}$, $K$ is a field,
and we assume that
$K$ contains a primitive $m^{\sth}$ root of unity. Thus,
$\mu_m(K)\cong \Z/m\Z$}.\\
\end{center}

\problem{9} Let $a\in K^\times$ and define:
\[
r = \text{Min}\{\ell : a^\ell = z^m,\text{ for some } z\in K^\times\}.
\]
Note that $r$ divides $m$. Let $B = \{z\in K^\times : z^{m/r}=a\}$.\\

\begin{enumerate}
\item Prove that for every $b\in B$, $x^r-b\in K[x]$ is irreducible.\\

\item Show that $\ds x^m-a = \prod_{b\in B} x^r-b$.\\

\item Let $L/K$ be the splitting extension of $x^m-a$. 
Prove that $\Gal{L}{K}$ is cyclic of size $r$.\\
\end{enumerate}

\problem{10} Let $G$ be a cyclic group of size $m$. Show that
we have a (not natural) isomorphism:
\[
\Hom_{\scriptstyle{\text{gp}}}(G,\mu_m(K))
\cong G.
\]
(Hint: an isomorphism can be written upon choosing 
a primitive $m^{\sth}$ root of unity $\zeta\in \mu_m(K)$
and a generator $\sigma$ of $G$.)

\end{document}
