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\def\Z{{\mathbb Z}}
\def\Q{{\mathbb Q}}
\def\R{{\mathbb R}}
\def\lra{\longrightarrow}

\centerline{\bf Mathematics 361:  Number Theory}
\centerline{\bf Assignment F}

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{\bf Reading:  }Ireland and Rosen, Chapter 6 (including the exercises)
and into Chapter~7

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{\bf Problems:}

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Ireland and Rosen, Exercises 6.2; 6.4, 6.5;
%6.6 (don't bother proving that $\Q[\alpha]$ is a ring but rather show
%that this ring is in fact a field, i.e., the ring $\Q[\alpha]$ of
%polynomials in~$\alpha$ is all of the field $\Q(\alpha)$ of rational
%functions in~$\alpha$), 6.7;
6.11, 6.12, 6.13, 6.14; 6.23, 6.18; 6.19 (note that
$\cos(2\pi/5)=(\zeta_5+\zeta_5^{-1})/2$ where $\zeta_5=e^{2\pi i/5}$
and that the polynomial of~$\zeta_5$ is $1+X+X^2+X^3+X^4$).

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Some of these exercises may be facilitated by invoking or first
establishing the following two results:
\begin{itemize}
\item Given a nonzero polynomial~$f$ with rational coefficients, there
  exists a unique positive rational number $r$ such that $rf$ has
  integral coefficients with greatest common divisor~$1$.
\item Granting the result of exercise~6.4, if a nonzero integral
  polynomial $g(X)\in\Z[X]$ factors in $\Q[X]$ then it factors in~$\Z[X]$.  
\end{itemize}

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