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\begin{document}

\begin{center}
{\bf Math 100C: Spring 2017 \\ Homework 5 \\ Due 5/17/2017}
\end{center}





\vspace{0.1in}
\noindent
{\bf Problem 1:}  
Let $F$ be a field of characteristic zero and let $f \in F[x]$ be a monic polynomial with derivative $f'$.
Let $g \in F[x]$ be an {\em irreducible} polynomial with $g \mid f$ and $g \mid f'$.  
Prove that $g^2 \mid f$.

\vspace{0.1in}
\noindent
{\bf Problem 2:}  
Let $p$ be a prime, let $r$ be a positive integer, and let $q = p^r$.  To what finite
abelian group is the additive group
$\mathbb{F}_q^+$ of the $q$-element field $\mathbb{F}_q$ isomorphic?

\vspace{0.1in}
\noindent
{\bf Problem 3:} 
Factor the polynomials $x^9 - x$ and $x^{27} - x$ over the field $\FF_3$.



\vspace{0.1in}
\noindent
{\bf Problem 4:}  
Let $p$ be a prime number, let $r \geq 1$, and let $q = p^r$.  Let $K$ be a field of order $q$.  Define 
$\sigma: K \rightarrow K$ by $\sigma(x) = x^p$.  Prove the following:
\begin{enumerate}
\item  $\sigma: K \rightarrow K$ is a field automorphism.
\item  $\sigma$ has order $r$.
\item  $\sigma$ generates the group $\mathrm{Aut}(K)$ of field automorphisms of $K$.
\end{enumerate}

\vspace{0.1in}
\noindent
{\bf Problem 5:}
Prove that every finite extension of a finite field has a primitive element.

\vspace{0.1in}
\noindent
{\bf Problem 6:}
Let $F$ be a finite field and let $f(x) \in F[x]$ be a nonconstant polynomial with $f'(x) = 0$.  Prove that 
$f(x)$ is not irreducible in $F[x]$.

\vspace{0.1in}
\noindent
{\bf Problem 7:}
The following polynomials are symmetric in the variables $x_1, x_2, x_3$.  Write them in terms of the
elementary symmetric polynomials $s_1, s_2, s_3$.
\begin{align*}
f(x) &=  x_1^3 + x_2^3 + x_3^3 \\
g(x) &= x_1^2 x_2 + x_1 x_2^2 + x_1^2 x_3 + x_1 x_3^2 + x_2^2 x_3 + x_2 x_3^2 
\end{align*}


\vspace{0.1in}
\noindent
{\bf Problem 8:} 
(Optional - not to be handed in.)
Is there an analog of character theory for $G$-sets?  
Let $G$ be a finite group and let $S$ and $T$ be finite $G$-sets.
We say that $S$ and $T$ are {\em isomorphic} as $G$-sets if there is a bijection $\phi: S \rightarrow T$ such that 
$\phi(g.s) = g.\phi(s)$ for all $g \in G$ and $s \in S$.
\begin{enumerate}
\item  Given $s \in S$, let $G_s := \{g \in G \,:\, g.s = s \}$ be the stabilizer of $S$.  Prove that for any $g \in G$ and $s \in S$
we have $G_{g.s} = g G_s g^{-1}$.
\item  Prove that every orbit of the action of $G$ on $S$ is isomorphic to the action of $G$ on the set of left cosets $G/H$
given by $g(g'H) := (gg')H$, for some subgroup $H \subseteq G$.
\item  Prove that $S$ and $T$ are isomorphic as $G$-sets if and only if the following condition holds:
\begin{quote}
for any subgroup $H \subseteq G$ which is the stabilizer of some element of $S$ or of $T$, we have $|S^H| = |T^H|$,
where $S^H = \{ s \in S \,:\, h.s = s$ for all $h \in H\}$ and similarly for $T^H$.
\end{quote}
\end{enumerate}
This criterion for determining $G$-set isomorphism is equivalent to the {\em Frobenius table of marks}.  Explain
why it is impractical for $G = S_n$ in general.
The table of marks is not often used.

\vspace{0.1in}
\noindent
{\bf Problem 9:} 
(Optional - not to be handed in.)
{\bf Restriction of representations.}
Let $G$ be a finite group and let $H$ be a subgroup of $G$.  If 
$R: G \rightarrow GL_n(k)$ is a matrix representation of $G$, the 
{\em restriction of $R$ to $H$} is the function
$$
R \downarrow_H: H \rightarrow GL_n(k)
$$
obtained by restricting $R$ to $H \subseteq G$.
\begin{enumerate}
\item  If $V$ is a $G$-module, what is the restriction $V \downarrow_H$ (i.e., how can we think of it)?
\item  It $\chi: G \rightarrow k$ is the character of a matrix representation $R$ of $G$, what is the 
character $\chi \downarrow_H: H \rightarrow k$ of $R \downarrow_H$?
\item  Show that if $V$ is an irreducible $G$-module, then $V \downarrow_H$ is not necessarily
an irreducible $H$-module.
\end{enumerate}


\vspace{0.1in}
\noindent
{\bf Problem 10:} 
(Optional - not to be handed in.)
{\bf Induction of representations.}
Let $G$ be a finite group and let $G \subseteq H$ be a subgroup of index $[G:H] = r$.
Let $R: H \rightarrow GL_n(k)$ be a representation of $H$.  We define a representation 
$R \uparrow^G: G \rightarrow GL_{nr}(k)$ 
(the {\em induction of $R$ to $G$}) as follows.  Let $T = \{t_1, t_2, \dots, t_r \}$ be a left transversal
for $H$ in $G$, so that every element  $g \in G$ may be written uniquely as $g = t_i h$ for some $1 \leq i \leq r$ 
and $h \in H$.  Given $g \in G$, we define $R \uparrow^G(g)$ to be the block matrix
$$
R\uparrow^G(g) := 
\begin{pmatrix}
R(t_1^{-1} g t_1) & \cdots & R(t_r^{-1} g t_1) \\
 & \vdots & \\
 R(t_r^{-1} g t_1) & \cdots & R(t_r^{-1} g t_r) 
\end{pmatrix},
$$
where we interpret $R(t_i^{-1} g t_j) = 0$ if $t_i^{-1} g t_j \notin H$.

\begin{enumerate}
\item  Prove that for any $g \in G$, the matrix $R \uparrow^G(g)$ is a `block permutation matrix' -- just one of the blocks
in every row and column above is nonzero.   Also show that $R \uparrow^G(1)$ is the identity matrix in $GL_{nr}(k)$.
\item  Prove that for any $x, y \in G$ we have $R \uparrow^G(xy) = R \uparrow^G(x) R \uparrow^G(y)$, so that
$R \uparrow^G$ is actually a representation of $G$.
\item  Let $\chi: H \rightarrow k$ be the character of $R$.  Prove that the character $\chi \uparrow^G$ of $R \uparrow^G$ 
is given by the formula
$$
\chi \uparrow^G(g) = \sum_{i = 1}^r \chi(t_i^{-1} g t_i),
$$
where we make the convention that $\chi(x) = 0$ if $x \notin H$.
\item  Now assume $k = \CC$.  Prove that $\chi \uparrow^G$ is also given by the formula
$$
\chi \uparrow^G(g) = \frac{1}{|H|} \sum_{x \in G} \chi( x g x^{-1}).
$$
Deduce that the representation $R \uparrow^G$ does not depend on the choice of the transversal $T$, up to isomorphism.
\item  Give an example of an irreducible matrix representation $R$ of $H$ over $\CC$ such that $R \uparrow^G$ is not 
irreducible.
\end{enumerate}



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