Math 581f Midterm

The midterm as a pdf: midterm.pdf

Here's the raw source, in case you are latexing your solutions:

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\Large\bf Take Home Midterm for Math 581F\\
Due Monday November 5, 2007, at the start of class.
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There are five problems.  Do not talk to other people about the
problems, though you may email me if you have questions about problems
(in case they are ill-stated).  You may use Sage if you very
explicitly describe exactly what calculations you did, and you may
consult any book or web page (cite it).  Make sure to do a good job on
this exam, since it is {\em worth 30\% of your grade}, almost the same
as your entire homework.  And it's not a terribly hard exam either.  I
fully expect everbody to get a perfect score.

\begin{enumerate}
\item
\begin{enumerate}
\item Consider the map $\vphi: \Z^3 \to \Z^4$ given
by 
$$\vphi((1,0,0)) = (1,2,3,0), \quad \vphi((0,1,0)) = (4,5,6,0),\quad
\vphi((0,0,1)) = (7,8,9,0).$$ 
 Write the cokernel of
$\vphi$, i.e., $\Z^4/\Im(\vphi)$ as a direct sum of
cyclic groups. 
\item Which of the following rings are Noetherian?
\begin{enumerate}
\item $\Z[\pi, e, \sqrt{2}]$    -- adjoin some real numbers to $\Z$,
\item $\F_7(x_1,x_2,x_3)[y_1, y_2]$   -- adjoin some variables to a finite field,
\item $\Z[x_1, x_2, \ldots, x_n, \ldots, n\geq 1]/(x_1-x_2)$ -- adjoin infinitely
many variables to $\Z$ and quotient out by the ideal generated by $x_1 - x_2$,
\item $\overline{\Z}[x]$  -- adjoin a variable to the ring of all algebraic integers in a fixed choice of algebraic closure of $\QQ$.
\end{enumerate}

\end{enumerate}

\item 
\begin{enumerate}
\item Prove that if $K$ is a number field then there
are infinitely many prime ideals of $\O_K$.
\item Suppose $I$ and $J$ are fractional ideals of
a number field $K$.
\begin{enumerate}
\item Prove that $I+J$ is a fractional ideal.
\item Prove that if $I=\prod \mathfrak{p}_i^{e_i}$ and
$J= \prod \mathfrak{p}_i^{f_i}$, then 
$I +J = \prod_{i=1}^r \mathfrak{p}_i^{\min(e_i, f_i)}$.
\end{enumerate}

\end{enumerate}

\item Let $\O_K$ be the ring of integers of the number
field $K=\Q(\sqrt{3^{997} - 1})$. [Note: You will probably not
get anywhere trying to create this number field directly
in the current version Sage.]
\begin{enumerate}
\item Find an order $R$ such that $[\O_K:R]$ is not divisible
by $7$. 
\item Give generators for the prime ideal factors of the
ideal $7\O_K$.
\end{enumerate}

\item Let $K=\Q(\sqrt{-23})$, and set $a=\sqrt{-23}\in K$..  Show how
to find $n \in \ZZ$
and $\alpha \in \O_K$, such that
$$
  \left( 2a + 2, a - 11, -\frac{5}{2}a - \frac{17}{2}\right) \O_K = (n, \alpha)\O_K.
$$
Explain the steps you use to find $n$ and $\alpha$ -- don't
just ``ask Sage''. 

\item Make a list of every integral ideal $I$ of the ring of integers
  of $K=\Q(\sqrt{-23})$ that has norm at most $10$.

\end{enumerate}

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