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%\twelvepoint
%\hbox to \hsize {\hbox to 0cm{{\bf Comput.\ Abstract Algebra}\hss}\hfill
%{\sl Take-Home Final Exam}\hfill
%\hbox to 0cm {\hss{\rm Spring 1990}}}
\tenpoint
\leftline{{\bf Comput.\ Abstract Agebra}}
\leftline{{\sl Take-Home Final Exam}}
\leftline{{\rm Spring 1990}}
\vskip 0.5truecm
\par\noindent{\sl Your Name:} \hbox to 5cm{\hrulefill}
\vskip 0.5truecm
This examination consists of 4 questions, that can be answered using
the {\sl Mathematica} packages provided in class.
Please submit not only the numeric answers, but also give an
explanation on the way you computed them.
\par
If you decide to submit your actual computation (although a hand-written
explanation suffices), you can prepare an input file,
say {\tt probX.m} (see the {\tt*.m} files in the {\tt Algebra/Notebooks} directory
and run it as ``{\tt math < probX.m > probX.pr}''.
You then can print the output file by typing ``{\tt lpr -Pslw probX.pr}'';
it will print on the laserwriter at the back of the Sun-Lab.
Make sure you load the {\tt init.m}
file in the {\tt Notebooks} directory to get the input lines echoed.
\par
You are not allowed to consult a classmate for help with the problems, but
you can ask J.-M. Wiley for help. The exam is due Wednesday, May~9, at noon.
Please return your exam to J.-M., or to either Donna or Ora in Amos Eaton~119.
\hfill Good Luck!
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$$\vbox{\halign{\hfill#\hskip 0.25cm&#\hbox to 1cm{\hrulefill}\cr
Problem 1&\cr
2&\cr
3&\cr
4&\cr
\omit\cr
Total&\cr}}$$
\medskip
\noindent{\bf Problem 1} (5 points):
Find an integer solution in $x, y$, and $z$ to
$$
  295927\;x + 301337\;y + 104759\;z = 1.
$$
\medskip\noindent
\noindent{\bf Problem 2} (5 points):
Find an integer solution in $x$ and $y$ to
$$ 10^{25} + 13 = x^2 + y^2.$$
[Hint: See Homework~3 and {\tt Factor.m}.]
\medskip\noindent
{\bf Problem 3} (5 points):
Find the first odd prime $p$ such that the polynomial
$$
   \displaylines{
   \quad 3721041 - 1486824 {x^2} + 1440748 {x^4} - 582456 {x^6} \hfill\cr
   \hfill+ 154038 {x^8}  - 22552 {x^{10}} + 1932 {x^{12}} - 72 {x^{14}} + {x^{16}}\quad\cr}
$$
has all linear factors when factored modulo~$p$.
You can generate this polynomial with the package {\tt Polynom.m} by
typing
$$
  \hbox{\tt Polynom`SwinnertonDyer[x, 2, \ttlb-1, 2, 3, 5\ttrb]}.
$$
\medskip\noindent
{\bf Problem 4} (5 points): How many complex roots
with both non-negative real and non-negative imaginary part, i.e.,
roots in the first quadrant, does the polynomial
$$
1 + x + 5 {x^2} + 4 {x^3} + 10 {x^4} + 6 {x^5} + 11 {x^6} + 7 {x^7} +
   9 {x^8} + 4 {x^9} + 2 {x^{10}} - {x^{11}}
$$
have?
\bye