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\begin{document}
\section*{Lior Silberman's Math 501: Problem Set 5 (due 16/10/2020)}
\begin{center}
\textbf{Finite fields}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] (The Frobenius map) Let $K$ be a field of characteristic
$p>0$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that the map $x\mapsto x^{p}$ defines a monomorphism
$K\to K$ fixing the prime field.
\item [{(b)}] Conclude by induction that the same holds for the map $x\mapsto x^{p^{r}}$
for any $r\geq1$.
\item [{(c)}] When $K$ is finite show that the Frobenius map is an automorphism.
\item [{({*}d)}] When $K$ is an arbitrary algebraic extension of $\FF_{p}$
show that the Frobenius map is again an automorphism.
\item [{FACT}] We obtain a group homomorphism $\Z\to\Gal(\bar{\FF}_{p}:\FF_{p})$.
We will later show that the image of this homomorphism is dense.\bigskip{}
\end{lyxlist}
\item [{2.}] (Multiplicative groups)
\begin{lyxlist}{10.}
\item [{(a)}] Let $G$ be a finite $p$-group such that for every $d$,
$\left|\left\{ g\in G\mid g^{d}=e\right\} \right|\leq d$. Show that
$G$ is cyclic.
\item [{(b)}] Let $G$ be a finite group such that for every $d$, $\left|\left\{ g\in G\mid g^{d}=e\right\} \right|\leq d$.
Show that $G$ is cyclic.
\item [{({*}c)}] Let $F$ be a field, $G\subset F^{\times}$ a finite multiplicative
subgroup. Show that $G$ is cyclic.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{3.}] (Uniqueness of finite fields) Fix a prime $p$ and let $q=p^{r}$
for some $r\geq1$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that the polynomial $x^{q}-x\in\FF_{p}[x]$ is separable.
\item [{(b)}] Let $F$ be a finite field with $q$ elements. Show that
$F$ is a splitting field for $x^{q}-x$ over $\FF_{p}$.
\item [{(c)}] Conclude that for each $q$ there is at most one isomorphism
class of fields of order $q$. If such a field exists it is denoted
$\FF_{q}$.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{4.}] (Existence of finite fields) Fix a prime $p$ and let $q=p^{r}$
for some $r\geq1$.
\begin{lyxlist}{10.}
\item [{(a)}] Let $F/\FF_{p}$ be a splitting field for $x^{q}-x$, and
let $\sigma\colon F\to F$ be the map $\sigma(x)=x^{q}$. Show that
the polynomial also splits in the fixed field of $\sigma$.
\item [{(b)}] Conclude that the field $F$ has order $q$. \bigskip{}
\end{lyxlist}
\item [{5.}] Let $F$ be a finite field, $K/F$ a finite extension.
\begin{lyxlist}{10.}
\item [{(a)}] Show that the extension $K/F$ is normal and separable.
\item [{(b)}] Show that the extension is \emph{simple}: there exists $\alpha\in K$
so that $K=F(\alpha)$.
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Simple extensions}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{{*}6.}] Let $K(\alpha):K$ be a simple extension.
\begin{lyxlist}{10.}
\item [{(a)}] If $\alpha$ is algebraic, show that there are finitely many
subfields $M$ of $K(\alpha)$ containing $K$.\\
\emph{Hint:} consider the minimal polynomial of $\alpha$ over $M$.
\item [{(b)}] If $\alpha$ is transcendental, show that there are infinitely
many intermediate fields $M$.\bigskip{}
\end{lyxlist}
\item [{7.}] Let $L:K$ be an extension of fields with finitely many intermediate
subfields.
\begin{lyxlist}{10.}
\item [{(a)}] Show that the extension is algebraic.
\item [{(b)}] Show that the extension is \emph{finitely generated}: there
exists a finite subset $S\subset L$ so that $L=K(S)$.
\item [{(c)}] Show that the extension is finite.\bigskip{}
\end{lyxlist}
\item [{{*}{*}8.}] Let $L:K$ be an extension of infinite fields with finitely
many intermediate fields. Show that it is a simple algebraic extension.
\begin{lyxlist}{10.}
\item [{RMK}] We will later show that every separable extension satisfies
the hypothesis. For finite fields see 5(b).\bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{center}
\textbf{Supplementary problem: Algebraicity and algebraic closures}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Let $L:K$ be an extension of fields
\begin{lyxlist}{10.}
\item [{(a)}] Let $\alpha\in L$. Show that $\alpha$ is algebraic if and
only if there a subset $E\subset L$ such that: (1) $E$ is a subspace
of $L$, thought of as a $K$-vectorspace; (2) $\alpha E\subset E$.
\item [{(b)}] Obtain a new proof of Corollary \ref{cor:algebraic-elements}
as follows: if $\alpha,\beta\in L$ stabilize $E,F\subset L$ respectively,
then the required elements stabilizer an image of $E\otimes_{K}F$
in $L$, which is necessarily finite-dimensional.\bigskip{}
\end{lyxlist}
\item [{B.}] Let $M:L$ and $L:K$ be algebraic extensions of fields. Show
that $M:K$ is algebraic.
\end{lyxlist}
\begin{defn*}
A field extension $K\embed\bar{K}$ is called an \emph{algebraic closure}
if it is algebraic, and if every polynomial in $K[x]$ splits in $\bar{K}[x]$.
We also say informally that $\bar{K}$ is an \emph{algebraic closure
of }$K$.
\end{defn*}
\begin{lyxlist}{10.}
\item [{RMK}] The following problems depend on basic notions from set theory:
cardinality and Zorn's Lemma.
\item [{C.}] Let $K\embed L$ be an algebraic extension.
\begin{lyxlist}{10.}
\item [{(a)}] If $K$ is finite, show that $\left|L\right|\leq\aleph_{0}$.
\item [{(b)}] If $K$ is infinite, show that $\left|L\right|=\left|K\right|$.\bigskip{}
\end{lyxlist}
\item [{D.}] Let $K\embed\bar{K}$ be an algebraic closure. Show that every
algebraic extension of $\bar{K}$ is an isomorphism of fields.\bigskip{}
\item [{E.}] (Existence of algebraic closures) Let $K$ be a field, $X$
an infinite set containing $K$ with $\left|X\right|>\left|K\right|$.
Let $0,1$ denote these elements of $K\subset X$. Let
\[
\mathcal{F}=\left\{ \left(L,+,\cdot\right)\mid K\subset L\subset X,\,\left(L,0,1,+,\cdot\right)\textrm{ is a field with }K\subset L\textrm{ an algebraic extension}\right\} \,.
\]
Note that we are assuming that restricting $+,\cdot$ to $K$ gives
the field operations of $K$.\bigskip{}
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\mathcal{F}$ is a set. Note that $\left\{ (\varphi,L)\mid L\textrm{ is a field and }\varphi\colon K\to L\textrm{ is an algebraic extension}\right\} $
is not a set.
\item [{(b)}] Show that every algebraic extension of $K$ is isomorphic
to an element of $\mathcal{F}$.
\item [{(c)}] Given $\left(L,+,\cdot\right)$ and $\left(L',+',\cdot'\right)\in\mathcal{F}$
say that $\left(L,+,\cdot\right)\leq\left(L',+',\cdot'\right)$ if
$L\subseteq L'$, $+\subseteq+'$, $\cdot\subseteq\cdot'$. Show that
this is a transitive relation.
\item [{(d)}] Let $\bar{K}\in\mathcal{F}$ be maximal with respect to this
order. Show that $\bar{K}$ is an algebraic closure of $K$.
\item [{(e)}] Show that $K$ has algebraic closures.
\item [{\bigskip{}
}]~
\end{lyxlist}
\item [{F.}] (Uniqueness of algebraic closures) Let $K\embed\bar{K}$ and
$K\embed L$ be two algebraic closures of $K$. Show that the two
extensions are isomorphic.\\
\emph{Hint: }Let $\mathcal{G}$ be the set of $K$-embeddings intermediate
subfields $K\subset M\subset L$ into $\bar{K}$, ordered by inclusion.\bigskip{}
\end{lyxlist}
\end{document}