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\title{Exercises for Sections 5 and 6:\\Signature and Roots of Unity}
\author{Math 582e, Winter 2009, University of Washington}
\date{\bf Due Wednesday February 18, 2009}
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\begin{enumerate}
\item Let $f = x^{5} - 3 x^{4} + 3 x^{3} - 7 x + 6$.
\begin{enumerate}
\item Find any {\em choice} of polynomials $A_0, A_1,\ldots, A_k \in
\RR[x]$ as in our algorithm from class (Section~4.1.2 of Cohen GTM
138) for computing the number of real roots of $f$.
\item Explicitly find the set $Z$ of all real zeros of all these polynomials.
\item For each $x\in Z$, compute the sign sequence $A_0(x), A_1(x), \ldots,
A_k(x)$.
\item How many real roots does $f$ have?
\end{enumerate}
\item (Exercise 25 of Chapter 4 of Cohen GTM 138) Let $\alpha$ be an
algebraic integer of degree $d$ all of whose conjugates have absolute
value $1$.
\begin{enumerate}
\item Show that for every positive integer $k$, the monic minimal
polynomial of $\alpha^k$ in $\ZZ[X]$ has all of its coefficients
bounded in absolute value by $2^d$.
\item Deduce from this that there exists only a finite number of distinct
powers of $\alpha$, hence that $\alpha$ is a root of unity.
\end{enumerate}
\item (Exercise 26 of Chapter 4 of Cohen GTM 138) Let $\rho\in \OO_K$
be an algebraic integer given as a polynomial in $\theta$, where
$K=\QQ(\theta)$ and $T$ is the minimal monic polynomial of $\theta$
in $\ZZ[X]$. Give two algorithms to check exactly whether or not
$\rho$ is a root of unity, and compare their efficiency.
\item Show that the number field obtained by adjoing a root of $x^4+1$
to $\QQ$ contains exactly $8$ roots of unity. Do not simply run the Sage command {\tt K.number\_of\_roots\_of\_unity()}.
\end{enumerate}
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