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\title{Math 772\\Summer 2006\\Homework 2}
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\centerline{\it Due Monday July 3, 2006.}

\centerline{}

\noindent 1. (5 pt) Show that the equation $x^4+y^4=z^2$ has no nontrivial solutions for $x,y,z\in\mathbb{N}$. Hint: show that given a solution with $x,y,z\in\mathbb{N}$, show that a ``smaller" positive solution can be obtained. This technique is referred to as ``infinite descent". (Note that this problem takes care of the Fermat problem for $n=4$.)

\centerline{}

\noindent 2. (5 pt) Let $R$ be an integral domain with quotient field $K$. We define the {\it integral closure} of $R$ to be 

\[
\overline{R}=\{\alpha\in K\vert p(\alpha)=0\text{ for some monic }p(x)\in R[x]\}.
\]

\noindent We say that $R$ is integrally closed if $R=\overline{R}$ (that is, $R$ already contains all of its integral elements from $K$). Prove that any UFD is integrally closed.

\centerline{}

\noindent 3. (5 pt) Let $R$ be an integral domain with quotient field $K$ and $p\in R$ a nonzero prime element. Show that $p$ is also a prime element of $R[x]$.

\centerline{}

\noindent 4. (5 pt) Let $F$ be a field extension of degree $n$ over $\mathbb{Q}$. Suppose that $\omega\in F$ is a root of a monic polynomial in $\mathbb{Z}[x]$. Show that $\omega$ is a root of a monic polynomial in $\mathbb{Z}[x]$ of degree no more than $n$ and, in particular, show that the minimal polynomial of $\omega$ (over $\mathbb{Q}$) may be taken to be monic and in $\mathbb{Z}[x]$.

\centerline{}


\noindent 5. (5 pt) Let $d$ be a square-free integer. Show that the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{d})$ is given by

\[
R=\begin{cases}

\mathbb{Z}[\sqrt{d}] & \text{if $d\equiv 2,3\ \text{mod}(4)$,}\\
\mathbb{Z}[\frac{1+\sqrt{d}}{2}] & \text{if $d\equiv 1\ \text{mod}(4)$.}
\end{cases}
\]


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