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\begin{document}
\section*{Lior Silberman's Math 501: Problem Set 8 (due 13/11/2020)}
\begin{center}
\textbf{(From PS7) 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 [{1.}] 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 [{2.}] (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}
\begin{center}
\textbf{Examples}
\par\end{center}
\begin{lyxlist}{10.}
\item [{3.}] (Quadratic extension) Let $L=K(\sqrt{d})$ be a quadratic
extension of a field of characteristic not equal to $2$.
\begin{lyxlist}{10.}
\item [{(a)}] Write down the matrix of multiplication by $\alpha=a+b\sqrt{d}\in L$
in the basis $\left\{ 1,\sqrt{d}\right\} $.
\item [{(b)}] Find the trace and determinant of this matrix.
\item [{(c)}] Let $\sigma$ be the non-trivial element of $\Gal(L/K)$.
Show that the answers to (b) agree with $\alpha+\sigma(\alpha)$,
$\alpha\sigma(\alpha)$ respectively.
\item [{RMK}] Meditate on the case $L=\C$, $K=\R$.\bigskip{}
\end{lyxlist}
\item [{4.}] (Cyclotomic extension) Let $\zeta_{p}$ be a primitive root
of unity of order $p$ and equip $\Q(\zeta)$ with the basis $\left\{ 1,\zeta_{p},\ldots,\zeta_{p}^{p-2}\right\} $.
Let $G$ be the cyclic group $\Gal(\Q(\zeta_{p}):\Q)$.
\begin{lyxlist}{10.}
\item [{(a)}] Write down the matrix of multiplication by $\zeta_{p}$ in
this basis.
\item [{(b)}] Find the trace and determinant of this matrix.
\item [{({*}c)}] Find its characteristic polynomial.
\item [{({*}d)}] Explicitly compute $\sum_{\sigma\in G}\sigma(\zeta_{p})$
and $\prod_{\sigma\in G}\sigma(\zeta_{p})$ and show that they equal
your answers from parts (b),(d).\bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{center}
\textbf{The trace}\bigskip{}
\par\end{center}
When $L/K$ is a finite Galois extension and $\alpha\in L$ we encounter
in class the combination (``trace'') $\Tr_{K}^{L}(\alpha)=\sum_{\sigma\in\Gal(L/K)}\sigma\alpha$,
which we need to be non-zero. We will study this construction when
$L/K$ is a finite separable extension, fixed for the purpose of the
problems 3-5.\bigskip{}
\begin{lyxlist}{10.}
\item [{5.}] Let $N/K$ be a finite normal extension containing $L$.
\begin{lyxlist}{10.}
\item [{(a)}] For $\alpha\in L$ we provisionally set $\Tr_{K}^{L}(\alpha)=\sum_{\mu\colon L\to N}\mu\alpha$
(``trace of $\alpha$''), $N_{K}^{L}(\alpha)=\prod_{\mu\colon L\to N}\mu\alpha$
(``norm of $\alpha$''). Show that the definition is independent
of the choice of $N$.
\item [{(b)}] Making a judicious choice of $N$ show that the trace and
norm defined in part (a) are elements of $K$.
\item [{(c)}] Show that when $L/K$ is a Galois extension the definition
from part (a) reduces to the combination used in class.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{6.}] (Elements of zero trace) In class we had the occasion to need
elements $\alpha\in L$ with trace zero. For this, let $L_{0}=\left\{ \alpha\in L\mid\Tr_{K}^{L}(\alpha)=0\right\} $.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\Tr_{K}^{L}$ is a $K$-linear functional on $L$,
so that $L_{0}$ is a $K$-subspace of $L$.
\item [{(b)}] When $\chr(K)=0$, show that $L=K\oplus L_{0}$ as vector
spaces over $K$ (direct sum of vector spaces; the analogue of direct
product of groups). Conclude that when $\left[L:K\right]\geq2$ the
set $L_{0}\setminus K$ is non-empty. (e..g the normal closure).
\item [{(c)}] Show that $\Tr_{K}^{L}$ is a non-zero linear functional
in all characteristics.
\item [{(d)}] Show that $L_{0}$ is not contained in $K$ unless $[L:K]=\chr(K)=2$,
in which case $L_{0}=K$, or $[L:K]=1$ in which case $L_{0}=\left\{ 0\right\} $.
\bigskip{}
\end{lyxlist}
\item [{7.}] (Yet another definition) We continue with the separable extension
$L/K$ of degree $n$.
\begin{lyxlist}{10.}
\item [{(a)}] Let $f\in K[x]$ be the (monic) minimal polynomial of $\alpha\in L$,
say that $f=\sum_{i=0}^{d}a_{i}x^{i}$ with $a_{d}=1$. Show that
$\Tr_{K}^{K(\alpha)}(\alpha)=-a_{d-1}$ and that $N_{K}^{K(\alpha)}(\alpha)=(-1)^{d}a_{0}$.
\item [{(b)}] Show that $\Tr_{K}^{L}(\alpha)=-\frac{n}{d}a_{d-1}$ and
that $N_{K}^{L}(\alpha)=(-1)^{n}a_{0}^{n/d}$.\\
\emph{Hint:} Recall the proof that $[L:K]$ has $n$ embeddings into
a normal closure.
\item [{(c)}] Show that $\Tr_{K}^{L}(\alpha)$ and $N_{K}^{L}(\alpha)$
are, respectively, the trace and determinant of multiplication by
$\alpha$, thought of as a $K$-linear map $L\to L$.\\
\emph{Hint:} Show that we have $L\isom(K(\alpha))^{n/d}$ as $K(\alpha)$-vector
spaces,.
\item [{\bigskip{}
}]~
\end{lyxlist}
\end{lyxlist}
\begin{defn*}
From now on we define the trace and norm of $\alpha$ as in 5(c).
Note that this definition makes sense even if $L/K$ is not separable.
\bigskip{}
\end{defn*}
\begin{lyxlist}{10.}
\item [{8.}] (Transitivity) Let $K\subset L\subset M$ be a tower of finite
extensions. Show that
\begin{lyxlist}{10.}
\item [{(a)}] $\Tr_{K}^{M}=\Tr_{K}^{L}\circ\Tr_{L}^{M}$.
\item [{(b)}] $N_{K}^{M}=N_{K}^{L}\circ N_{L}^{M}$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{\newpage Supplementary problems}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] (Purely inseparable extension) Let $L/K$ be an purely inseparable
algebraic extension of fields of characteristic $p$.
\begin{lyxlist}{10.}
\item [{(a)}] For every $\alpha\in L$ show that there exists $r\geq0$
so that $\alpha^{p^{r}}\in K$. In fact, show that the minimal polynomial
of $\alpha$ is of the form $x^{p^{r}}-\alpha^{p^{r}}$.\\
\emph{Hint}: Consider the minimal polynomials of $\alpha$ and $\alpha^{p}$
\item [{(b)}] Conclude that when $\left[L:K\right]$ is finite it is a
power of $p$.
\item [{(c)}] When $\left[L:K\right]$ is finite show that $\Tr_{K}^{L}$
is identically zero.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{B.}] Let $L=\C(x)$ (the field of rational functions in variable)
and for $f\in L$ let $\left(\sigma(f)\right)(x)=f(\frac{1}{x})$,
$\left(\tau(f)\right)(x)=f(1-x)$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\sigma,\tau\in\Aut(L)$ and that $\sigma^{2}=\tau^{2}=1$.
\item [{(b)}] Show that $G=\left\langle \sigma,\tau\right\rangle $ is
a subgroup of order $6$ of $\Aut(L)$ and find its isomorphism class.
\item [{(c)}] Let $K=\Fix(G)$. Find this field explicitly.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\end{document}