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\begin{center} {\Large\bf Algebra V63.0349 \\ Assignment $\omega$} \\ Due Friday, May 8 in recitation \end{center}
\noi {\bf FINAL EXAM:} The Final Exam will be on May 18, 10-11:50,
in our usual classroom. Latest information about the exam, sample
problems, etc., will be given on the website.
\beq
The mystery, as well as the glory, of mathematics lies not so much in
the fact that abstract theories do turn out to be useful in solving
problems but in that wonder of wonders, the fact that a theory
meant for solving one type of problem is often the only way of solving
problems of entirely different kinds, problems for which the theory
was not intended. These coincidences occur so frequently that they
must belong to the essence of mathematics.
\\ -- Gian-Carlo Rota
\enq
\ben
\item Assume that {\em both} $p$ and $2^p-1$ are primes. ({\tt Comment:}
When this occurs $2^p-1$ is called a {\em Mersenne prime}.) Let
$f(x)$ be an irreducible polynomial over $Z_2$ of degree $p$.
\ben
\item Show $f(x)|(x^{2^p-1}-1)$.
\item Show for all $1\leq i < 2^p-1$ that $f(x)$ does not divide $x^i-1$.
\item Show that $x$ is a generator of $Z_2[x]/(f(x))$.
\een
\item Consider the linear recursion
$u_n = 4u_{n-1}-5u_{n-2}$
with initial conditions $u_0=1,u_1=1$.
\ben
\item Give the solution using complex numbers.
\item Rewrite the solution in terms of real exponential
and trigonometric functions.
\item Discuss the asymptotic nature of $u_n$ as $n\ra\infty$.
(It might help to work out the first ten or so terms!)
\een
\item Let $f(x)\in Z_p[x]$. Suppose $f'(x)=0$. (That is, the zero
polynomial. E.g., with $p=5$ and $f(x)=x^{25}+2x^5+2$.) Prove
that $f(x)=g(x)^p$ for some $g(x)\in Z_p[x]$. {\tt Remark:} This
has an important implication. Let $g(x)\in Z_p[x]$ be irreducible
and have degree $n$.
Then, as $g'(x)=0$ is not possible, $\gcd[g(x),g'(x)]=1$. Let
$K\supset Z_p$ be an extension field in which $g(x)$ splits
completely. Then $g(x)$ must have {\em distinct} roots
$\ah_1,\ldots,\ah_n\in K$.
\item
Let $F$ be a finite field and let $\ah,\beta$ be
distinct nonzero elements of $F$. Let $\ah$ have
order $r$ and let $\beta$ have order $s$.
(The order of an element $\gam$ is the least
positive $m$ with $\gam^m=1$.) Let $M=lcm(r,s)$.
Let $a,b$ be nonzero elements of $F$ and define
a sequence $u_n=a\ah^n+b\beta^n$
\ben
\item Show $u_{n+M}=u_n$ for all $n$.
\item (*) Show that $M$ is the period of the sequence $u_n$. That is,
show there is no $L$, $1\leq L < M$, with $u_{n+L}=u_n$ for all $n$.
\een
\item Let $f(x)\in Z[x]$, a polynomial of degree $n$. Suppose $f(x)$
has $n$ {\em distinct} roots $\ah_1,\ldots,\ah_n\in C$. Call a prime
$p$ bad (given $f$) if $f(x)$ has a multiple factor when considered
in $Z_p[x]$. (For example, $x^2+x-8$ is $(x+2)^2$ in $Z_3[x]$ and
$(x+6)^2$ in $Z_{11}[x]$.) Prove that there can be only a {\em finite}
(maybe zero) number of bad primes $p$.
\een
\beq
A foolish consistency is the hobgoblin of little minds, adored by
little statesman and philosophers and divines. With consistency a
great soul has simply nothing to do.
\\ -- Ralph Waldo Emerson
\enq
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