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\begin{document}
\section*{Lior Silberman's Math 412: Problem Set 1 (due 18/1/2023)}
Practice problems numbered M1 etc), any sub-parts marked ``PRAC''
(practice) ``SUPP'' (supplementary) and supplementary problems are
not for submission.\vspace{-0.4cm}
\begin{center}
\textbf{Practice problems}
\par\end{center}
\begin{lyxlist}{10.}
\item [{M1.}] Show that the map $f\colon\R^{3}\to\R$ given by $f(x,y,z)=x-2y+z$
is a linear map. Show that the maps $\left(x,y,z\right)\mapsto1$
and $\left(x,y,z\right)\mapsto x^{2}$ are not.
\item [{M2.}] Let $F$ be a field, $X$ a set. Carefully show that pointwise
addition and scalar multiplication endow the set $F^{X}$ of functions
from $X$ to $F$ with the structure of an $F$-vectorspace.
\item [{M3.}] For $x\in X$ let $\delta_{x}\colon F^{X}\to F$ be evaluation
at $x$: $\delta_{x}(f)\eqdef f(x)$. Show that each $\delta_{x}$
is a linear map. Meditate on the fact that the vector space structure
was defined exactly so that $\delta_{x}$ are linear maps.\bigskip{}
\end{lyxlist}
\begin{center}
\textbf{For submission}
\par\end{center}
\begin{lyxlist}{10.}
\item [{RMK}] This problem introduces a device for showing that sets of
vectors are linearly independent. Make sure you understand how this
argument works. \bigskip{}
\item [{1.}] (Vector space axioms) Let $E=\left(E,0_{E},1_{E},+_{E},\cdot_{E}\right)$
be a field, and let $F\subset E$ be a subfield, in other words a
subset which is closed under the operations and satsifies the field
axioms. It is commonly said that $E$ naturally has the structure
of an $F$-vectorspace.
\begin{lyxlist}{10.}
\item [{(a)}] Explicitely write the quadruple that was implicitely introduced
in the statement of the problems.
\item [{(b)}] Verify the vector space axioms for this putative vector space
structure.\\
(\emph{Hint}: this problem is tedious rather than hard)\bigskip{}
\end{lyxlist}
\item [{2.}] Let $V$ be a vector space, $S\subset V$ a set of vectors.
A \emph{minimal dependence} in $S$ is an equality $\sum_{i=1}^{m}a_{i}\vv_{i}=\zv$
where $\vv_{i}\in S$ are distinct, $a_{i}$ are scalars not all of
which are zero, and $m\geq1$ is as small as possible so that such
$\left\{ a_{i}\right\} $, $\left\{ \vv_{i}\right\} $ exist.
\begin{lyxlist}{10.}
\item [{---}] It is implicit in the following that either $S$ is independent
or it has a minimal dependence. Make this explicit in your mind (don't
write this bit up).
\item [{PRAC}] Find a minimal dependence among $\left\{ \begin{pmatrix}1\\
1\\
0
\end{pmatrix},\begin{pmatrix}1\\
0\\
1
\end{pmatrix},\begin{pmatrix}1\\
1\\
1
\end{pmatrix},\begin{pmatrix}2\\
1\\
1
\end{pmatrix}\right\} \subset\R^{3}$.
\item [{(a)}] Show that in a minimal dependence the $a_{i}$ are all non-zero.
\item [{(b)}] Suppose that $\sum_{i=1}^{m}a_{i}\vv_{i}$ and $\sum_{i=1}^{m}b_{i}\vv_{i}$
are minimal dependences in $S$, involving the exact same set of vectors.
Show that there is a non-zero scalar $c$ such that $a_{i}=cb_{i}$.
\item [{(c)}] Let $T\colon V\to V$ be a linear map, and let $S\subset V$
be a set of (non-zero) eigenvectors of $T$, each corresponding to
a distinct eigenvalue. Applying $T$ to a minimal dependence in $S$
obtain a contradiction to (c) and conclude that $S$ is actually linearly
independent.
\item [{({*}d)}] Let $\Gamma$ be a group. The set $\Hom\left(\Gamma,\C^{\times}\right)$
of group homomorphisms from $\Gamma$ to the multiplicative group
of nonzero complex numbers is called the set of \emph{quasicharacters}
of $\Gamma$ (the notion of ``character of a group'' has an additional,
different but related meaning, which is not at issue in this problem).
Show that $\Hom\left(\Gamma,\C^{\times}\right)$ is linearly independent
in the space $\C^{\Gamma}$ of functions from $\Gamma$ to $\C$.
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{lyxlist}{10.}
\item [{SUPP}] In the setting of problem 1, a \emph{field homomorphism}
is a map $\sigma\colon E\to E$ respecting the field operations. This
map is an \emph{automorphism} if it is invertible, equivalently if
it is 1-1 and onto. Let $\Aut(E)$ be the set of field automorphisms
of $E$, and let
\[
\Aut_{F}(E)=\left\{ \sigma\in\Aut(E)\mid\sigma\restriction_{F}=\id_{F}\right\} \,.
\]
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\Aut(E)$ is a group and that $\Aut_{F}(E)$ is
a subgroup.
\item [{(b)}] Show that $\Aut(E)\subset E^{E}$ is linearly independent.\\
(\emph{Hint}: See 2(d)).
\item [{(c)}] Thinking of $E$ as an $F$-vectorspace as in problem 1,
show that $\Aut_{F}(E)\subset\Hom_{F}(E,E)$. Use part (b) to show
that $\Aut_{F}(E)$ is linearly independent, and conclude that $\#\Aut_{F}(E)\leq\left(\dim_{F}E\right)^{2}$
if $\dim_{F}E<\infty$.
\item [{RMK}] The integer $\left[E:F\right]=\dim_{F}E$ is usually called
the \emph{degree} of the field extension $E/F$. With more careful
work it is possible to obtain the bound $\#\Aut_{F}(E)\leq\left[E:F\right]$.\bigskip{}
\end{lyxlist}
\item [{3.}] (Matrices associated to linear maps) Let $V,W$ be vector
spaces of dimensions $n,m$ respectively. Let $T\in\Hom(V,W)$ be
a linear map from $V$ to $W$. Show that there are ordered bases
$B=\left\{ \vv_{j}\right\} _{j=1}^{n}\subset V$ and $C=\left\{ \vw_{i}\right\} _{i=1}^{m}\subset W$
and an integer $d\leq\min\left\{ n,m\right\} $ such that the matrix
$A=\left(a_{ij}\right)$ of $T$ with respect to those bases satisfies
$a_{ij}=\begin{cases}
1 & i=j\leq d\\
0 & \textrm{otherwise}
\end{cases}$, that is has the form
\[
\begin{pmatrix}1\\
& \ddots\\
& & 1\\
& & & 0\\
& & & & \ddots\\
& & & & & 0
\end{pmatrix}
\]
(Hint1: study some examples, such as the matrices $\begin{pmatrix} & 1\\
1
\end{pmatrix}$ and $\begin{pmatrix}2 & -4\\
1 & -2
\end{pmatrix}$) (Hint2: start your solution by choosing a basis for the image of
$T$).\bigskip{}
\end{lyxlist}
\begin{center}
\textbf{Extra credit: Finite fields}
\par\end{center}
\begin{lyxlist}{10.}
\item [{4.}] Let $F$ be a field.
\begin{lyxlist}{10.}
\item [{(a)}] Define a map $\iota\colon\left(\Z,+\right)\to\left(F,+\right)$
by mapping $n\in\Z_{\geq0}$ to the sum $1_{F}+\cdots+1_{F}$ $n$
times. Show that this extends to a ring homomorphism.
\item [{DEF}] If the map $\iota$ is injective we say that $F$ is of \emph{characteristic
zero}.
\item [{(b)}] Suppose there is a non-zero $n\in\Z$ in the kernel of $\iota$.
Show that the smallest positive such number is a prime number $p$.
\item [{DEF}] In that case we say that $F$ is of \emph{characteristic
$p$.}
\item [{(c)}] Show that in that case $\iota$ induces an isomorphism between
the finite field $\FF_{p}=\Z/p\Z$ and a subfield of $F$. In particular,
there is a unique field of $p$ elements up to isomorphism. \bigskip{}
\end{lyxlist}
\item [{5.}] Let $F$ be a field with finitely many elements. Show that
there exists an integer $r\geq1$ such that $F$ has $p^{r}$ elements.\\
(Hint: see problem 1)
\begin{lyxlist}{10.}
\item [{RMK}] For every prime power $q=p^{r}$ there is a field $\FF_{q}$
with $q$ elements, and two such fields are isomorphic. They are usually
called \emph{finite fields}, but also \emph{Galois fields} after their
discoverer.
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Supplementary Problems I: A new field }\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{A.}] Let $\Q(\sqrt{2})$ denote the set $\left\{ a+b\sqrt{2}\mid a,b\in\Q\right\} \subset\R$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\Q(\sqrt{2})$ is a $\Q$-subspace of $\R$.
\item [{(b)}] Show that $\Q(\sqrt{2})$ is two-dimensional as a $\Q$-vector
space. In fact, identify a basis.
\item [{({*}c)}] Show that $\Q(\sqrt{2})$ is a field.
\item [{({*}{*}d)}] Let $V$ be a vector space over $\Q(\sqrt{2})$ and
suppose that $\dim_{\Q(\sqrt{2})}V=d$. Show that $\dim_{\Q}V=2d$.
\end{lyxlist}
\end{lyxlist}
\newpage{}
\begin{center}
\textbf{Supplementary Problems II: How physicists define vectors }\bigskip{}
\par\end{center}
Fix a field $F$.
\begin{lyxlist}{10.}
\item [{B.}] (The general linear group)
\begin{lyxlist}{10.}
\item [{(a)}] Let $\GL_{n}(F)$ denote the set of invertible $n\times n$
matrices with coefficients in $F$. Show that $\GL_{n}(F)$ forms
a group with the operation of matrix multiplication.
\item [{(b)}] For a vector space $V$ over $F$ let $\GL(V)$ denote the
set of invertible linear maps from $V$ to itself. Show that $\GL(V)$
forms a group with the operation of composition.
\item [{(c)}] Suppose that $\dim_{F}V=n$ Show that $\GL_{n}(F)\isom\GL(V)$
(hint: show that each of the two group is isomorphic to $\GL(F^{n})$.
\end{lyxlist}
\item [{C.}] (Group actions) Let $G$ be a group, $X$ a set. An \emph{action}
of $G$ on $X$ is a map $\cdot\colon G\times X\to X$ such that $g\cdot(h\cdot x)=(gh)\cdot x$
and $1_{G}\cdot x=x$ for all $g,h\in G$ and $x\in X$ ($1_{G}$
is the identity element of $G$).
\begin{lyxlist}{10.}
\item [{(a)}] Show that matrix-vector multiplication $\left(g,\vv\right)\mapsto g\vv$
defines an action of $G=\GL_{n}(F)$ on $X=F^{n}$.
\item [{(b)}] Let $V$ be an $n$-dimensional vector space over $F$, and
let $\cB$ be the set of ordered bases of $V$. For $g\in\GL_{n}(F)$
and $B=\left\{ \vv_{i}\right\} _{i=1}^{\dim V}\in\cB$ set $gB=\left\{ \sum_{j=1}^{n}g_{ij}\vv_{i}\right\} _{j=1}^{n}$.
Check that $gB\in\cB$ and that $\left(g,B\right)\mapsto gB$ is an
action of $\GL_{n}(F)$ on $\cB$.
\item [{(c)}] Show that the action is \emph{transitive}: for any $B,B'\in\cB$
there is $g\in\GL_{n}(F)$ such that $gB=B'$.
\item [{(d)}] Show that the action is \emph{simply transitive}: that the
$g$ from part (b) is unique.
\end{lyxlist}
\item [{D.}] (From the physics department) Let $V$ be an $n$-dimensional
vector space, and let $\cB$ be its set of bases. Given $\vu\in V$
define a map $\phi_{\vu}\colon\cB\to F^{n}$ by setting $\phi_{\vu}(B)=\va$
if $B=\left\{ \vv_{i}\right\} _{i=1}^{n}$ and $\vu=\sum_{i=1}^{n}a_{i}\vv_{i}$.
\begin{lyxlist}{10.}
\item [{(a)}] Show that $\alpha\phi_{\vu}+\phi_{\vu'}=\phi_{\alpha\vu+\vu'}$.
Conclude that the set $\left\{ \phi_{\vu}\right\} _{\vu\in V}$ forms
a vector space over $F$.
\item [{(b)}] Show that the map $\phi_{\vu}\colon\cB\to F^{n}$ is \emph{equivariant}
for the actions of B(a),B(b), in that for each $g\in\GL_{n}(F)$,
$B\in\cB$, $g\left(\phi_{\vu}(B)\right)=\phi_{\vu}(gB)$.
\item [{(c)}] Physicists define a ``covariant vector'' to be an equivariant
map $\phi\colon\cB\to F^{n}$. Let $\Phi$ be the set of covariant
vectors. Show that the map $\vu\mapsto\phi_{\vu}$ defines an isomorphism
$V\to\Phi$. (Hint: define a map $\Phi\to V$ by fixing a basis $B=\left\{ \vv_{i}\right\} _{i=1}^{n}$
and mapping $\phi\mapsto\sum_{i=1}^{n}a_{i}\vv_{i}$ if $\phi(B)=\underline{a}$).
\item [{(d)}] Physicists define a ``contravariant vector'' to be a map
$\phi\colon\cB\to F^{n}$ such that $\phi(gB)=\leftidx^{t}g^{-1}\cdot\left(\phi(B)\right)$.
Verify that $\left(g,\va\right)\mapsto\leftidx^{t}g^{-1}\va$ defines
an action of $\GL_{n}(F)$ on $F^{n}$, that the set $\Phi'$ of contravariant
vectors is a vector space, and that it is naturally isomorphic to
the dual vector space $V'$ of $V$.
\end{lyxlist}
\end{lyxlist}
\begin{center}
\textbf{Supplementary Problems III: Fun in positive characteristic
}\bigskip{}
\par\end{center}
\begin{lyxlist}{10.}
\item [{E.}] Let $F$ be a field of characteristic $2$ (that is, $1_{F}+1_{F}=0_{F}$).
\begin{lyxlist}{10.}
\item [{(a)}] Show that for all $x,y\in F$ we have $x+x=0_{F}$ and $\left(x+y\right)^{2}=x^{2}+y^{2}$.
\item [{(b)}] Considering $F$ as a vector space over $\FF_{2}$ as in
problem 5, show that the field automorphism $\Frob\colon F\to F$
given by $\Frob(x)=x^{2}$ is an $\FF_{2}$-linear map.
\item [{(c)}] Suppose that the map $x\mapsto x^{2}$ is actually $F$-linear
and not only $\FF_{2}$-linear. Show that $F=\FF_{2}$.
\item [{RMK}] Compare your answer with practice problem 1.
\end{lyxlist}
\item [{F.}] (This problem requires a bit of number theory) Now let $F$
have characteristic $p>0$. Show that the \emph{Frobenius endomorphism
}$x\mapsto x^{p}$ is $\FF_{p}$-linear.
\end{lyxlist}
\end{document}