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\begin{center}
  {\large Mathematics 111}\\Spring 2007\\Homework 4
\end{center}

\begin{enumerate}
  % \item Justify that we can view the tensor products $\C\otimes_\R
  %   \C$ and $\C \otimes_\C \C$ both as (left) $\R$-modules and
  %   $\C$-modules discussing appropriate bi-module structure.
  %   Compute (and justify) their dimensions as real and complex
  %   vector spaces (four computations).

\item (Extending the base).  Let $\varphi: A \to B$ be a ring
  homomorphism.  We can view $B$ as a left and right $A$-module via
  $a\cdot b := \varphi(a) b$ and $b\cdot a := b\varphi(a)$.  A
  standard example is where $A \subset B$ and $\varphi$ is inclusion.
  So we can view $B$ as an $A-A$ bi-module.  Let $N$ be a left
  $A$-module.
  \begin{enumerate}
  \item Show that the abelian group $B\otimes_A N$ has a left
    $A$-module structure and that the map $N \to B\otimes_A N$ given
    by $n \mapsto 1_B\otimes n$ is $A$-linear.  Note that $B\otimes_A
    N$ has a natural left $B$-module structure, so it is natural to
    wonder to what extent $B\otimes_A N$ is a $B$-module
    ``containing'' the original $A$-module $N$. Not always as you will
    show, but it is true in many important cases.

  \item First the bad news.  Give an example where $\varphi$ is
    injective, but the associated map $N \to B\otimes_A N$ is not.

  \item Now suppose that $N$ is a free $A$-module with basis
    $\{n_i\}_{i\in I}$.  From class we know that $B\otimes_A N$ is a
    free left $B$-module with basis $\{1_B\otimes n_i\}_{i\in I}$.
    Under the assumption $N$ is free, show that if $\varphi$ is
    injective, so too is the map $N \to B\otimes_A N$.  In particular,
    if $L/K$ is a field extension and $V$ is a vector space over $K$,
    then $L \otimes_K V$ is a vector space over $L$ ``containing'' $V$
    and having the same dimension as $V$ over $K$.

  \item As an example of the above, show that if $L/K$ is a field
    extension and $V = K[x]$ is the polynomial ring over $K$, there is
    a natural isomorphism (as $L$-modules) between $L[x]$ and $L
    \otimes_K K[x]$.

  \item The previous isomorphism may be more subtle than it appears.
    Let $K$ be a field and $\alpha$ an element in its algebraic
    closure.  Despite the natural $L$-linear isomorphism between
    $L[x]$ and $L \otimes_K K[x]$, show that there are choices of $L$
    and $\alpha$ so that $L[\alpha]$ and $L\otimes_K K[\alpha]$ are
    not isomorphic as $L$-modules.
  \end{enumerate}

\item Some applications.  Let $K$ be a field, $f$ an irreducible
  polynomial in $K[x]$, and $\alpha$ a root of $f$ in some splitting
  field.
  \begin{enumerate}
  \item Show that for any field extension $L/K$, $L \otimes_K
    K(\alpha)$ is isomorphic (as an $L$-module) to the quotient
    $L[x]/(f(x))$.

  \item Let $K=\Q$, $\alpha=\root6\of 2$, $L = \Q(\root3\of 6)$.  $L
    \otimes_K K(\alpha)$ is isomorphic to a direct sum of fields (each
    an extension of $L$).  Find them.

  \item Let $L$ be a splitting field for a separable polynomial $f$
    over $K$, $\alpha \in L$ a root of $f$.  Show that $L \otimes_K
    K(\alpha)$ is isomorphic to $L^n$ where $n$ is the degree of $f$.
  \end{enumerate}
\end{enumerate}






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