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\begin{center} {\Large\bf Algebra V63.0344 \\ Assignment 7} \\ Due,
Friday, March 28 in Recitation \end{center}
\beq
I am slow to learn and slow to forget that which I have learned. My
mind is like a piece of steel; very hard to scratch anything on it
and almost impossible after you get it there to rub it out.
\\ Abraham Lincoln
\enq
\ben
\item Let $p$ be a prime of the form $p=3k+1$. Let $g$ be a generator, so
that the elements of $Z_p^*$ can be written $1,g,g^2,\ldots,g^{3k-1}$ with
$g^{3k}=1$.
\ben
\item Using this generator show that there is an element (actually, two elements) $\omega\in Z_p^*$ with $\omega\neq 1$ and
$\omega^3=1$.
\item \label{one} (*) Show that there exists $\eta\in Z_p^*$ with $\eta^2=-3$. [Idea: In $C$ write $\sqrt{-3}$ in terms
of $\omega=e^{2\pi i/3}$. Use this to make an ``inspired guess" for $\eta$ in terms of $\omega$.]
\een
\item Now suppose $p$ be a prime of the form $p=3k+2$.
\ben
\item Again using a
generator $g$ show that there is no
element $\omega\in Z_p^*$ with $\omega\neq 1$ and $\omega^3=1$.
\item (*) Show that there does not exist $\eta\in Z_p$ with $\eta^2=-3$.
[Idea: Reversing Problem \ref{one} find $\omega$ in terms of $\eta$.]
\een
\item \label{two} Let $F$ be a finite field with $q=p^n$ elements. Define
$\sig: F\ra F$ by $\sig(\ah)=\ah^p$.
\ben
\item Show that $\sig(\ah\beta)=\sig(\ah)\sig(\beta)$ for all $\ah,\beta\in F$.
\item Show that $\sig(\ah +\beta)=\sig(\ah)+ \sig(\beta)$ for all $\ah,\beta\in F$.
\item Show that $\sig(\ah^{-1})=\sig(\ah)^{-1}$ for all nonzero $\ah\in F$.
\item (*) Show that $\sig$ is injective. That is, show that if $\sig(\ah)=\sig(\beta)$ then $\ah=\beta$.
\item Deduce that $\sig$ is surjective.
\een
{\tt Note:} Together this gives that $\sig$
is an automorphism, an isomorphism from $F$ to itself. $\sig$
is called the Frobenius automorphism.]
\pagebreak
\item Let $F=Z_3[x]/(x^2+1)$.
\ben
\item List the elements of $F$.
\item Find a generator $g$ of $F^*$. (Some grunt work here.)
\item For each $\ah\in F$ find the minimal polynomial $p_{\ah}(y)$
of $\ah$ in $Z_3[y]$. (E.g., for $\ah=x$ take $p(y)=y^2+1$ as
then $p(x)=x^2+1=0$.) (Lots of grunt work here.)
\item Factor $y^9-y$ in $F[y]$.
\item Factor $y^9-y$ in $Z_3[y]$. Show how the factors in $F[y]$ join
to form factors in $Z_3[y]$.
\een
BTW, Note that when $\ah\in F$ is a root of a minimal polynomial
then so is $\sig(\ah)$ where $\sig$ is the Frobenius Automorphism
of Problem \ref{two}. For example, with $\ah=x$ the minimal
polynomial is $y^2+1$ and $\sig(\ah)=x^3=2x$ is another root of
$p(x)$.
\item Assume the following theorem: Let $q=p^n$ and set $f(x)=x^q-x$.
Then, in $Z_p[x]$, $f(x)$ factors into the product of all monic irreducible
(over $Z_p[x]$) polynomials of all degrees $d$, where $d$ is a divisor
(including $1$ and $n$) of $n$.
\ben
\item How many irreducible quadratic polynomials are there over $Z_5[x]$?
(Count degrees in the factorization of $x^{25}-x$.)
\item How many irreducible cubic polynomials are there over $Z_5[x]$?
(Count degrees in the factorization of $x^{125}-x$.)
\item (*)
How many irreducible polynomials of degree six are there over $Z_5[x]$?
(Count degrees in the factorization of $x^{15625}-x$, noting $15625=5^6$.)
\een
\een
\beq
I have never let my schooling interfere with my education. \\ -- Mark Twain
\enq
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