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\begin{document}

\begin{center}
{\bf Math 100C: Spring 2017 \\ Homework 6 \\ Due 5/24/2017}
\end{center}





\vspace{0.1in}
\noindent
{\bf Problem 1:}  
Let $F$ be a field and let $F[x_1, \dots, x_n]^{S_n}$ be the set of symmetric polynomials
with coefficients in $F$ over the variable set $\{x_1, \dots, x_n\}$.  We may regard 
$F[x_1, \dots, x_n]^{S_n}$ as an infinite-dimensional $F$-vector space.  Find two different 
$F$-bases for $F[x_1, \dots, x_n]^{S_n}$.

\vspace{0.1in}
\noindent
{\bf Problem 2:}  
Determine the splitting fields of the following polynomials over $\QQ$:
$$f(x) = x^3 - 2,  \, \, \, g(x) = x^4 - 1, \, \, \,  h(x) = x^4 + 1.$$

\vspace{0.1in}
\noindent
{\bf Problem 3:} 
Let $F$ be a field and let $f(x) \in F[x]$ be a monic polynomial of degree $n$.
Let $K$ be a splitting field for $f$ over $F$.  Prove that the degree $[K:F]$ divides $n!$.



\vspace{0.1in}
\noindent
{\bf Problem 4:}  
Let $\alpha = 2^{1/3}$ be the real cube root of $2$ and let $\omega = \exp(2 \pi i/3)$.  
Determine all automorphisms of the fields $\QQ(\alpha)$ and $\QQ(\alpha,\omega)$.

\vspace{0.1in}
\noindent
{\bf Problem 5:}
For each of the following sets of automorphisms of the field $\CC(t)$, determine the group of
automorphisms $G$ which they generate and describe the fixed field $\CC(t)^G$ explicitly.
\begin{enumerate}
\item  $\sigma(t) = t^{-1}$,
\item $ \sigma(t) = it, $
\item  $ \sigma(t) = -t, \tau(t) = t^{-1},$
\item $ \sigma(t) = \omega t, \tau(t) = t^{-1}$,  where $\omega = \exp(2 \pi i /3)$.
\end{enumerate}

\vspace{0.1in}
\noindent
{\bf Problem 6:}
Let $f(x) = (x^2 - 2x - 1)(x^2 - 2x - 7) \in \QQ[x]$.  Let $K$ be the splitting field for $f(x)$ over $\QQ$.
Determine all automorphisms of $K$.



\vspace{0.1in}
\noindent
{\bf Problem 7:}  (Optional - not to be handed in.)
{\bf Tensor products I.}
Let $R$ be a ring (not necessarily commutative), let $M$ be a right $R$-module,
\footnote{That is, we have a map $M \times R \rightarrow M$ such that $m \cdot 1 = m$,
$m \cdot (r_1 r_2) = (m \cdot r_1) \cdot r_2$, and $(m_1 + m_2) \cdot r = m_1 \cdot r + m_2 \cdot r$ 
always.} 
and let $N$ be a left $R$-module.  Let $\ZZ[M \times N]$ be the free abelian group with basis given by the {\em set}
$M \times N$.   The {\em tensor product}  $M \otimes_R N$ is the abelian group 
$$
M \otimes_R N := \ZZ[M \times N]/I,
$$
where $I$ is the subgroup of $\ZZ[M \times N]$ generated by all elements of the form 
\begin{align*}
&(m \cdot r, n) - (m, r \cdot n) \\
&(m + m', n) - (m, n) - (m', n) \\
&(m, n + n') - (m,n) - (m,n')
\end{align*}
for $m, m' \in M, n, n' \in N,$ and $r \in R$.  
Let $m \otimes n$ be the image of $(m,n)$ in $M \otimes_R N$ for $m \in M$ and $n \in N$.
We have {\em by construction} the Universal Property of the Tensor Product:
\begin{quote}
for any abelian group $A$ and any map of sets $\phi: M \times N \rightarrow A$ which satisfies 
$\phi(m \cdot r, n) = \phi(m, r \cdot n),
\phi(m + m', n) = \phi(m,n) + \phi(m',n),$ and $\phi(m, n+n') = \phi(m,n) + \phi(m,n')$ always, we have
 a unique well-defined homomorphism of abelian groups $\Phi: M \otimes_R N \rightarrow A$ determined by
 $$\Phi: m \otimes n \mapsto \phi(m,n).$$
\end{quote}
Suppose that $S$ is another ring and $M$ is an $(S,R)$-bimodule.  
\footnote{That is, $M$ is a left $S$-module, a right $R$-module, and we have $s \cdot (m \cdot r) = r \cdot (m \cdot s)$
for all $s \in S, m \in M, r \in R$.}
Show that the map
$S \times (M \otimes_R N) \rightarrow (M \otimes_R N)$ determined by $s \cdot (m \otimes n) := (s \cdot m) \otimes n$
gives $M \otimes_R N$ the (well defined) structure of a left $S$-module.



\vspace{0.1in}
\noindent
{\bf Problem 8:} 
(Optional - not to be handed in.)
{\bf Tensor Products II.}  Let $K/F$ be a field extension and let $V$ be an $F$-vector space.  Explain how the 
tensor product $K \otimes_F V$ is a $K$-vector space.

Let $V$ and $W$ be vector spaces over some field $F$.  Explain how the tensor product $V \otimes_F W$ is an
$F$-vector space.

\vspace{0.1in}
\noindent
{\bf Problem 9:} 
(Optional - not to be handed in.)
{\bf Tensor Products III.}
Let $F$ be a field and let $V$ and $W$ be $F$-algebras.  Prove that the tensor product $V \otimes_F W$ has 
an $F$-algebra structure determined by
$$
(v \otimes w) \cdot (v' \otimes w') := (v \cdot v') \otimes (w \cdot w')
$$
for all $v, v' \in V$ and $w, w' \in W$.  To what familiar $F$-algebra is the tensor product
$F[x] \otimes_F F[y]$ isomorphic (where $x$ and $y$ are variables)?


\vspace{0.1in}
\noindent
{\bf Problem 10:} 
(Optional - not to be handed in.)
{\bf Tensor Products IV.}  Let $G$ and $H$ be groups and let $F$ be a field.  Let $V$ be a $G$-module over $F$ and let 
$W$ be an $H$-module over $F$.  Prove that the $F$-vector space $V \otimes_F W$ is a $G \times H$-module
according to the rule
$$
(g,h).(v \otimes w) := (g.v) \otimes (h.w)
$$
for all $g \in G, h \in H, v \in V, w \in W$.


\



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