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\begin{document}
\section*{Lior Silberman's Math 501: Problem Set 7 (due 6/11/2020)}
\begin{center}
\textbf{Galois theory}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] Let $L/K$ be a finite Galois extension. Let $K\subset M_{1},M_{2}\subset L$
be two intermediate fields. Show that the following are equivalent:
\begin{enumerate}
\item $M_{1}/K$ and $M_{2}/K$ are isomorphic extensions.
\item There exists $\sigma\in\Gal(L:K)$ such that $\sigma(M_{1})=M_{2}$.
\item $\Gal(L:M_{i})$ are conjugate subgroups of $\Gal(L:K)$.\bigskip{}
\end{enumerate}
\item [{2.}] ($V$-extensions) Let $K$ have characteristic different from
$2$.
\begin{lyxlist}{10.}
\item [{(a)}] Suppose $L/K$ is normal, separable, with Galois group $C_{2}\times C_{2}$.
Show that $L=K(\alpha,\beta)$ with $\alpha^{2},\beta^{2}\in K$.
\item [{(b)}] Suppose $a,b\in K$ are such that none of $a,b,ab$ is a
square in $K$. Show that $\Gal(K(\sqrt{a},\sqrt{b}):K)\isom C_{2}\times C_{2}$.\bigskip{}
\end{lyxlist}
\item [{3.}] (The generalized quaternion group). Let $G$ be a non-commutative
group of order $8$. Show that either $G\isom D_{8}=C_{2}\ltimes C_{4}$
or $G\isom Q_{8}=\left\langle i,j,k\mid i^{2}=j^{2}=k^{2},\,i^{4}=1,\,ij=k,\,ji=i^{2}k\right\rangle $
(the elememt $i^{2}=j^{2}=k^{2}$ is usually denoted $-1$ so the
elements of the group are $\left\{ \pm1,\pm i,\pm j,\pm k\right\} $.
\bigskip{}
\item [{{*}{*}4.}] Let $\alpha=\sqrt{(2+\sqrt{2})(3+\sqrt{3})}.$
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\left[\Q(\alpha):\Q\right]=8$ and that this extension
is normal.
\item [{(b)}] Show that $\Gal(\Q(\alpha):\Q)\isom Q_{8}$. \bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{The fundamental theorem of algebra}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{5.}] (Preliminaries)
\begin{lyxlist}{10.}
\item [{(a)}] Show that every finite extension of $\R$ has even order.
\item [{(b)}] Show that every quadratic extension of $\R$ is isomorphic
to $\C$.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{6.}] (Punch-line)
\begin{lyxlist}{10.}
\item [{(a)}] Let $F:\R$ be a finite extension. Show that $[F:\R]$ is
a power of $2$.\\
\emph{Hint}: Consider the $2$-Sylow subgroup of the Galois group
of the normal closure.
\item [{(b)}] Show that every proper algebraic extension of $\R$ contains
$\C$.
\item [{(c)}] Show that every proper extension of $\C$ contains a quadratic
extension of $\C$.
\item [{(d)}] Show that $\C:\R$ is an algebraic closure.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{center}
\textbf{Example: Cyclotomic fields}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{PRAC}] For practice (but not for submission)
\begin{lyxlist}{10.}
\item [{(a)}] Show that $x^{n}-1\in\Q[x]$ has $n$ distinct roots.
\item [{(b)}] Write $\mu_{n}$ for the set of roots of this polynomial.
Show that it forms a cyclic group of order $n$.
\item [{DEF}] $\mu_{n}$ is called the \emph{group of roots of unity of
order {[}dividing{]} $n$. }A root of unity $\zeta\in\mu_{n}$ is
called \emph{primitive} if it is a generator, that is if it has order
exactly $n$. We write $\zeta_{n}$ for a primitive root of unity
of order $n$, for example $e^{\frac{2\pi i}{n}}\in\C$ (by problem
6(a) the choice doesn't matter). For the purpose of the problem set
we also write $P_{n}\subset\mu_{n}$ for the set of primitive roots
of unity of order $n$. The polynomial $\Phi_{n}(x)=\prod_{\zeta\in P_{n}}(x-\zeta)$
is called the $n$th \emph{cyclotomic polynomial}. The field $\Q(\zeta_{n})$
is called the $n$th \emph{cyclotomic field}.
\item [{(c)}] Show that $\prod_{d|n}\Phi_{d}(x)=x^{n}-1$. We'll later
show that this is the factorization of $x^{n}-1$ into irreducibles
in $\Q[x]$.\bigskip{}
\end{lyxlist}
\item [{6.}] Let $\zeta_{n}$ be a primitive $n$th root of unity.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\Q(\zeta_{n})$ is the splitting field of $x^{n}-1$
over $\Q$.
\item [{(b)}] Let $G=\Gal(\Q(\zeta_{n}):\Q)$. For $\sigma\in G$ show
there is a unique $j\in\left(\Z/n\Z\right)^{\times}$ so that $\sigma(\zeta_{n})=\zeta_{n}^{j(\sigma)}$
and that $j\colon G\to(\Z/n\Z)^{\times}$ is an injective homomorphism
(we'll later show that this map is an isomorphism).
\item [{(c)}] Show that $\Phi_{n}(x)\in\Q[x]$ and that the degree of $\Phi_{n}$
is exactly $\phi(n)=\#(\Z/n\Z)^{\times}$.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{7.}] (prime power and prime order) Fix an odd prime $p$ and let
$r\geq1$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\Phi_{p^{r}}(x)=\frac{x^{p^{r}}-1}{x^{p^{r-1}}-1}$
and that this polynomial is irreducible.
\item [{(b)}] Show that $\Gal(\Q(\zeta_{p^{r}}):\Q)\isom\left(\Z/p^{r}\Z\right)^{\times}$.
\item [{RMK}] Parts (a),(b) hold for $p=2$ as well.
\item [{(c)}] Show that $\Gal(\Q(\zeta_{p^{r}}):\Q)$ is cyclic.
\item [{(d)}] Show that $\Q(\zeta_{p})$ has a unique subfield $K$ so
that $[K:\Q]=2$.
\item [{(e)}] Let $G=\Gal(\Q(\zeta_{p}):\Q)$. Show that there is a unique
non-trivial homomorphism $\chi\colon G\to\left\{ \pm1\right\} $.
\item [{(f)}] Let $g=\sum_{\sigma\in G}\chi(\sigma)\sigma(\zeta_{p})$
(the ``Gauss sum''). Show that $g\in K$ and that $g^{2}\in\Q$.
\item [{({*}g)}] Show that $g^{2}=(-1)^{\frac{p-1}{2}}p$, hence that $K=\Q(g)$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\end{document}