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\begin{document}
\section*{Lior Silberman's Math 501: Problem Set 3 (due 2/10/2020)}
\begin{center}
\textbf{Fields and extensions}
\par\end{center}
\begin{lyxlist}{10.}
\item [{1.}] (Concrete extensions) By Eisenstein's criterion and Gauss's
Lemma, the polynomials $x^{2}-2,x^{3}-2\in\Q[x]$ are irreducible.
Without using tools from abstract algebra (except that a root of an
irreducible polynomial isn't also a root of a polynomial of smaller
degree):
\begin{lyxlist}{10.}
\item [{(a)}] Let $K=\Q(\alpha)$ where $\alpha^{2}=2$ (this means: ``$K$
is an extension of $\Q$ generated by an element $\alpha$ so that
$\alpha^{2}=2$''). Show that $\left\{ 1,\alpha\right\} \subset K$
are linearly independent over $\Q$.
\item [{(b)}] Show that $\left\{ 1,\alpha\right\} $ spans $K$ (hint:
you need to show that the span is a subfield of $K$; start by showing
it's a subring). Conclude that $\left[K:\Q\right]=2$.
\item [{(c)}] Repeat with appropriate modifications for $L=\Q(\beta)$
where $\beta^{3}=2$.\bigskip{}
\end{lyxlist}
\item [{2.}] (The hard way) Continuing with the notation of proble 1, let
$\gamma\in L$ satisfy $\gamma^{3}=2$.
\begin{lyxlist}{10.}
\item [{(a)}] Write $\gamma=a+b\beta+c\beta^{2}$ and convert the equation
$\gamma^{3}=2=2+0\beta+0\beta^{2}$ to a system of three non-linear
equations in the three variables $a,b,c$ (justify your claim!).
\item [{(b)}] Taking a clever linear combination of two of the equations,
show that $a=0$.
\item [{(c)}] Now show that $b=1$, $c=0$, that is that $\gamma=\beta$.\bigskip{}
\end{lyxlist}
\item [{3.}] (The easy way) Let $\gamma\in L$ satisfy $\gamma^{3}=2$
and suppose $\gamma\neq\beta$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\zeta=\gamma/\beta$ satisfies $\zeta^{3}=1$.
\item [{(b)}] Let $m\in\Q[x]$ be the minimal polynomial of $\zeta$ over
$\Q$. Show that $\deg m=2$.\\
\emph{Hint}: Start by showing that $m$ is an irreducible factor of
$x^{3}-1$.
\item [{(c)}] Consider the field $\Q(\zeta)\subset L$. Show that $[\Q(\zeta):\Q]=2$
and obtain a contradiction.\\
\emph{Hint:} $[L:\Q]=[L:\Q(\zeta)]\cdot[\Q(\zeta):\Q]$.\bigskip{}
\end{lyxlist}
\item [{{*}4.}] Let $K=\Q(\sqrt{2}+\sqrt{3})\subset L=\Q(\sqrt{2},\sqrt{3})$.
Show that $K=L$.\bigskip{}
\end{lyxlist}
\begin{center}
\textbf{Fields of fractions}
\par\end{center}
\begin{lyxlist}{10.}
\item [{5.}] Let $R$ be an integral domain.
\begin{lyxlist}{10.}
\item [{(a)}] Consider the set $X$ of \emph{formal expresions} $\frac{a}{b}$
where $a,b\in R$ and $b\neq0$. Define a relation on $X$ by $\frac{a}{b}\sim\frac{c}{d}$
if $ad=bc$. Show that this is an equivalence relation.
\item [{(b)}] Let $F$ be the set $X/\sim$ of equivalence relations, and
define operations on $F$ by $\left[\frac{a}{b}\right]+\left[\frac{c}{d}\right]=\left[\frac{ad+bc}{bd}\right]$,
$\left[\frac{a}{b}\right]\cdot\left[\frac{c}{d}\right]=\left[\frac{ac}{bd}\right]$.
Show that these are well-defined and give $F$ the structure of a
ring.
\item [{(c)}] Show that $F$ is a field, and that $a\mapsto\left[\frac{a}{1}\right]$
defines an embedding $\iota\colon R\to F$.
\item [{DEF}] $F$ is called the \emph{field of fractions} of $R$. If
$K$ is a field, the field of fractions of the polynomial ring $K[x]$
is called the \emph{field of rational functions} (in one variable)
\emph{over }$K$ and denoted $K(x)$.
\item [{(d)}] Show that for any field $K$, any injective ring homomorphism
$\iota\colon R\to K$ extends uniquely to a homomorphism $F\to K$
compatible with $\iota$.\bigskip{}
\end{lyxlist}
\item [{6.}] Fix an extension of fields $\iota\colon K\to L$.
\begin{lyxlist}{10.}
\item [{(a)}] Carefully show that for any $\alpha\in L$ there is a unique
homomorphism of rings $\psi_{\alpha}\colon K[x]\to L$ (``evaluation
at $\alpha$'') restricting to $\iota$ on $K$ and satisfying $\psi_{\alpha}(x)=\alpha$.
\item [{(b)}] Suppose that $\alpha$ is transcendental over $K$. Show
that $\iota$ extends uniquely to a map $\tilde{\psi}_{\alpha}\colon K(x)\to L$
so that $\tilde{\psi}_{\alpha}(x)=\tilde{\psi}_{\alpha}(\frac{x}{1})=\alpha$.
\item [{\bigskip{}
}]~
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Supplement I: the two quadratic $\R$}-algebras\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Let $i$ be a formal symbol, and let $\C$ be the set of formal
expressions $a+bi$ where $a,b\in\R$. Set $\left(a+bi\right)+\left(c+di\right)\eqdef(a+c)+(b+d)i$
and $\left(a+bi\right)\cdot\left(c+di\right)\eqdef(ac-bd)+(ad+bc)i$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that the definition makes $\C$ into a ring.
\item [{(b)}] Show that $\left\{ a+0i\mid a\in\R\right\} $ is a subfield
of $\C$ isomorphic to $\R$.
\item [{(c)}] Show that the \emph{complex conjugation }map $\tau(a+bi)=a-bi$
is a ring isomorphism $\tau\colon\C\to\C$ which restricts to the
identity map on the image of $\R$ from part (b).
\item [{(d)}] Show that for $z\in\C$ the condition $z\in\R$ and $\tau z=z$
are equivalent. Conclude that $Nz=N_{\R}^{\C}z\eqdef z\cdot\tau z$
is a multiplicative map $\C\to\R$.
\item [{(e)}] Show that $\C$ is a field.\emph{}\\
\emph{Hint}: Show first that if $z\in\C$ is non-zero then $Nz$ is
non-zero.\bigskip{}
\end{lyxlist}
\item [{B.}] Let $\R$ be the field of real numbers. Let $A=\left\{ a+bi\mid a,b\in\R\right\} $
where $i$ is a formal symbol, and define $(a+bi)+(c+di)\eqdef(a+b)+(c+d)i$,
$(a+bi)(c+di)\eqdef(ac+2bd)+(ad+bc)i$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that the definition makes $A$ into a ring.
\item [{(b)}] Show that $\left\{ a+0i\mid a\in\R\right\} $ is a subfield
of $A$ isomorphic to $\R$.
\item [{(c)}] Show that the \emph{complex conjugation }map $\tau(a+bi)=a-bi$
is a ring isomorphism $\tau\colon A\to A$ which restricts to the
identity map on the image of $\R$ from part (b).
\item [{(d)}] Show that for $z\in A$ the condition $z\in\R$ and $\tau z=z$
are equivalent. Conclude that $Nz=N_{\R}^{A}z\eqdef z\cdot\tau z$
is a multiplicative map $A\to\R$.
\item [{(e)}] Show that $A\isom\R\oplus\R$, and in particular that it
is not a field.
\item [{(f)}] Assume that multiplication is defined by $(a+bi)(c+di)\eqdef(ac+tbd)+(ad+bc)i$
for some fixed $t\in\R$. For which $t$ is the algebra a field? Find
the isomorphism class of the algebra, depending on $t$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Supplement II: More on Laurent series}\bigskip{}
\par\end{center}
\begin{defn*}
Let $R$ be a ring. A \emph{formal} \emph{Laurent series} over $R$
is a formal sum $f(x)=\sum_{i\geq i_{0}}a_{i}x^{i}$, in other words
a function $a\colon\Z\to R$ for which there exists $i_{0}\in\Z$
so that $a_{i}=0$ for all $i\leq i_{0}$. We define addition and
multiplication in the obvious way and write $R((x))$ for the set
of Laurent series. For non-zero $f\in R((x))$ let $v(f)=\min\left\{ i\mid a_{i}\neq0\right\} $
(``order of vanishing at $0$''; also set $v(0)=\infty$). Then
set $\left|f\right|=q^{-v(f)}$ ($\left|0\right|=0$) where $q>1$
is a fixed real number.
\end{defn*}
\begin{lyxlist}{10.}
\item [{C.}] (Invertibility)
\begin{lyxlist}{10.}
\item [{(a)}] Show that $1-x$ is invertible in $R[[x]].$\emph{}\\
\emph{Hint:} Find a candidate series for $\frac{1}{1-x}$ and calculate
the product.
\item [{(b)}] Show that $R[[x]]^{\times}=\left\{ a+xf\mid a\in R^{\times},\,f\in R[[x]]\right\} $.
\item [{(c)}] Show that $f\in R((x))$ is invertible iff it is non-zero
and $a_{v(f)}\in R^{\times}$.
\item [{(d)}] Show that $F((x))$ is a field for any field $F$.\bigskip{}
\end{lyxlist}
\item [{D.}] (Locality) Let $F$ be a field.
\begin{lyxlist}{10.}
\item [{(a)}] Let $I\ideal F[[x]]$ be a non-zero ideal. Show that $I=x^{n}F[[x]]$
for some $n\geq1$.\\
\emph{Hint}: Show that every nonzero $f\in F[[x]]$ can be uniquely
written in the form $x^{v(f)}g(x)$ where $g\in F[[x]]^{\times}$.
\item [{(b)}] Show that the natural map $F[x]/x^{n}F[x]\to F[[x]]/x^{n}F[[x]]$
is an isomorphism.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{lyxlist}{10.}
\item [{E.}] (Completeness)
\begin{lyxlist}{10.}
\item [{(a)}] Show that $v(fg)=v(f)+v(g)$, equivalently that $\left|fg\right|=\left|f\right|\left|g\right|$
for all $f,g\in R((x))$.
\item [{(b)}] Prove the \emph{ultrametric inequality} $v(f+g)\geq\min\left\{ v(f),v(g)\right\} \iff\left|f+g\right|\leq\max\left\{ \left|f\right|,\left|g\right|\right\} $
and conclude that $d(f,g)=\left|f-g\right|$ defines a metric on $f$.
\item [{(c)}] Show that $\left\{ f_{n}\right\} _{n=1}^{\infty}\subset R((x))$
is a Cauchy sequence iff there exists $i_{0}$ such that $v(f_{n})\geq i_{0}$
for all $n$, and if for each $i$ there exists $N=N(i)$ and $r\in R$
so that for $n\geq N$ the coefficient of $x^{i}$ in $f_{n}$ is
$r$.
\item [{(d)}] Show that $\left(R((x)),d\right)$ is complete metric space.
\item [{(e)}] Show that $R[[x]]$ is closed in $R((x))$.
\item [{(f)}] Show that $R[[x]]$ is compact iff $R$ is finite.\bigskip{}
\end{lyxlist}
\item [{F.}] (Ultrametric Analysis) Let $\left\{ a_{n}\right\} _{n=1}^{\infty}\subset R((x))$.
Show that $\sum_{n=1}^{\infty}a_{n}$ converges in $R((x))$ iff $\lim_{n\to\infty}a_{n}=0$.\\
\emph{Hint}: Assume first that $a_{n}\in R[[x]]$ for all $n$, and
for each $k$ consider the projection of $\sum_{n=1}^{N}a_{n}$ to
$R[[x]]/x^{k}R[[x]]$.\bigskip{}
\item [{G.}] (The degree valuation) Let $F$ be a field.
\begin{lyxlist}{10.}
\item [{(a)}] For $f\in F[x]$ set $v_{\infty}(f)=-\deg(f)$ (and set $v_{\infty}(0)=\infty)$.
Show that $v_{\infty}(fg)=v_{\infty}(f)+v_{\infty}(g)$. Show that
$v_{\infty}\left(f+g\right)\geq\min\left\{ v_{\infty}(f),v_{\infty}(g)\right\} $.
\item [{(b)}] Extend $v_{\infty}$ to the field $F(x)$ of rational functions
and show that it retains the properties above. For a rational function
$f$ you can think of $v_{\infty}(f)$ as ``the order of $f$ at
$\infty$'', just like $v(f)$ measures the order of $f$ at zero.
\item [{(c)}] Show that the completion of $F(x)$ \wrt the metric coming
from $v_{\infty}$ is exactly $R((\frac{1}{x}))$.\bigskip{}
\end{lyxlist}
\end{lyxlist}
\end{document}