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\title{Math 129: Algebraic Number Theory\\
Homework Assignment 10}
\author{William Stein}
\date{Due: Thursday, April 29, 2004}
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\maketitle

\begin{enumerate}
\item Prove that the ring $C$ defined in Section 9 really is the tensor
product of $A$ and $B$, i.e., that it satisfies the defining universal
mapping property for tensor products.  Part of this problem is for you
to look up a functorial definition of tensor product.

\item Find a zero divisor pair in $\Q(\sqrt{5})\tensor_\Q\Q(\sqrt{5})$.

\item 
\begin{enumerate}
\item Is $\Q(\sqrt{5})\tensor_\Q\Q(\sqrt{-5})$ a field?
\item Is $\Q(\sqrt[4]{5})\tensor_\Q\Q(\sqrt[4]{-5})\tensor_\Q\Q(\sqrt{-1})$ a field?
\end{enumerate}

\item Suppose $\zeta_5$ denotes a primitive $5$th root of unity.  For
  any prime $p$, consider the tensor product $\Q_p \tensor_\Q
  \Q(\zeta_5) = K_1\oplus \cdots \oplus K_{n(p)}$.  Find a simple
  formula for the number $n(p)$ of fields appearing in the
  decomposition of the tensor product $\Q_p \tensor_\Q \Q(\zeta_5)$.
  To get full credit on this problem your formula must be correct, but
  you do {\em not} have to prove that it is correct.

\item Suppose $\normspc_1$ and $\normspc_2$ are 
equivalent norms on a finite-dimensional vector space
$V$ over a field $K$ (with valuation $\absspc$). 
Carefully prove that the topology induced by $\normspc_1$
is the same as that induced by $\normspc_2$.

\item Suppose $K$ and $L$ are number fields (i.e., finite
extensions of $\Q$).  Is it possible for the tensor
product $K\tensor_\Q L$ to contain a nilpotent element? 
(A nonzero element $a$ in a ring $R$ is {\em nilpotent} if 
there exists $n>1$ such that $a^n=0$.)

\item  Let $K$ be the number field $\Q(\sqrt[5]{2})$.
\begin{enumerate}
\item In how many ways does the $2$-adic valuation $\absspc_2$ on $\Q$
extend to a valuation on $K$?
\item Let $v=\absspc$ be a valuation on $K$ that extends $\absspc_2$.
Let $K_v$ be the completion of $K$ with respect to $v$.
What is the residue class field $\F$ of $K_v$?
\end{enumerate}


\end{enumerate}


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