\documentclass[12pt]{article}
\usepackage{fullpage}
\usepackage{amsmath, amsthm, amssymb, paralist}

\usepackage[all]{xy}
\DeclareMathOperator{\Hom}{Hom}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}

\newcommand{\R}{{\mathbb R}}

\begin{document}
\thispagestyle{empty}

\begin{center}
  {\large Mathematics 111}\\Spring 2011\\Homework 4
\end{center}

\begin{enumerate}
\item (A theorem not proven in class).  Let $R$ be a ring with
  identity, $M$ a right $R$-module and $N$ a free left $R$-module with
  basis $\{e_i\}_{i\in I}$.  Show that every element in $M \otimes_R
  N$ can be written uniquely as a finite sum $\sum_{i\in I}m_i \otimes
  e_i$, $m_i \in M$.  Here uniquely means that if $\sum_i m_{i_k}\otimes e_{i_k}
  =\sum_i m'_{i_k}\otimes e_{i_k}$, then $m_{i_k} = m'_{i_k}$ for all
  $i_k$.

Hint:  Use the fact that tensor product commutes with the coproduct to
reduce to the one-dimensional case.  Then prove that as abelian groups
$M \otimes_R R e_i \cong M$ by exhibiting inverse homomorphisms
between them.



\item  Show that, as a corollary to the above problem, if $R \subset
  S$ are rings with $1_R = 1_S$, then for any free left $R$-module $N$
  with basis $\{e_i\}_{i \in I}$, $S\otimes_R N$ is a free left $S$-module
  with basis $\{1_S\otimes e_i\}_{i\in I}$.


\item (problem 25, D\&F, 10.4) Let $R \subset S$ be commutative rings
  with $1_R = 1_S$.  Show that as $S$-algebras, $S\otimes_R R[x] \cong
  S[x]$, where $x$ is an indeterminate over $S$.  Be sure to pay
  attention to justifying that your isomorphism is $S$-linear.

\item (problem 26, D\&F, 10.4) Let $R \subset S$ be commutative rings
  with $1_R = 1_S$, and let $I$ be an ideal in the polynomial ring
  $R[x_1,\dots,x_n]$.  Show that as $S$-algebras, \newline $S\otimes_R
  (R[x_1, \dots, x_n]/I) \cong S[x_1, \dots, x_n]/IS[x_1, \dots,
  x_n]$.

\item Let $L = \Q(\zeta_5)$ be the fifth cyclotomic field and let $K =
  \Q(\root 5\of2)$.  Using problem 4, determine $L \otimes_\Q K$ as an
  $L$-algebra.  It is a vector space over $L$, and its dimension can
  be computed in two ways one of which uses field theory.  Are your
  answers consistent?

\item Using problem 4, compute $\Q(\sqrt2) \otimes_\Q \Q(\root 4\of
  2)$ as a $\Q$-algebra in two ways (thinking of the tensor product as
  an extension of scalars in two ways).  Are your summands all
  isomorphic as $\Q$-algebras?  Why or why not?



\end{enumerate}






\end{document}
