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\title{Math 772\\Fall 2011\\Homework 4}
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\centerline{\it Due Friday, November 18, 2011.}

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\noindent 1. Find the class group for each quadratic ring of integers below.
\begin{itemize}
\item[a)] (5 pt) $\mathbb{Z}[\sqrt{-14}]$.
\item[b)] (5 pt) $\mathbb{Z}[\sqrt{-10}]$.
\item[c)] (5 pt) $\mathbb{Z}[\frac{1+\sqrt{-23}}{2}]$.
\item[d)] (5 pt) $\mathbb{Z}[\sqrt{-21}]$
\item[e)] (5 pt) $\mathbb{Z}[\frac{1+\sqrt{-163}}{2}]$.
\end{itemize}

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\noindent 2. (5 pt) Find the smallest positive square-free integer, $d$, such that the ring of integers of the field $\mathbb{Q}[\sqrt{d}]$ is not a UFD.

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\noindent 3. (5 pt) Explicitly show for quadratic fields that a prime is ramified if and only if it divides the discriminant.

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\noindent 4. Consider the field $\mathbb{Q}(\alpha)$ where $\alpha$ is a root of $x^5-x^3+1$. You may assume that the ring of integers of $\mathbb{Q}(\alpha)$ is $\mathbb{Z}[\alpha]$.
\begin{itemize}
\item[a)] (5 pt) Find the number of real and complex embeddings of $\mathbb{Q}(\alpha)$ into $\mathbb{C}$.
\item[b)] (5 pt) Find the discriminant of the field $\mathbb{Q}(\alpha)$.
\item[c)] (5 pt) Find the class group of the ring $\mathbb{Z}[\alpha]$.
\item[d)] (5 pt) Determine how the ramified primes factor in $\mathbb{Z}[\alpha]$.
\item[e)] (5 pt) Show that there is an element of norm 27 and of norm 9, but no element of norm 3.
\end{itemize}



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