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Math 581f Midterm

The midterm as a pdf: midterm.pdf

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


\Large\bf Take Home Midterm for Math 581F\\
Due Monday November 5, 2007, at the start of class.

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.

\item Consider the map $\vphi: \Z^3 \to \Z^4$ given
$$\vphi((1,0,0)) = (1,2,3,0), \quad \vphi((0,1,0)) = (4,5,6,0),\quad
\vphi((0,0,1)) = (7,8,9,0).$$ 
 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?
\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 $\QQ$.


\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 $K$.
\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 
$I +J = \prod_{i=1}^r \mathfrak{p}_i^{\min(e_i, f_i)}$.


\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.]
\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 $7\O_K$.

\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 $10$.



2013-05-11 18:33