\input amstex
\documentstyle{amsppt}
\loadbold
\TagsOnRight
\nologo
\nopagenumbers
\mag 1200
\topmatter
\title{Mat 567 Fall 2008, Number Theory I, Problems 8}\endtitle
\endtopmatter
\document
\centerline{\bf To be submitted by Tuesday, October 21st}
\medskip
\noindent {\bf Easier problems}
\settabs\+aaaa&aaaaa&aaaaaaaaaaaaa\cr
\medskip
\noindent 1. Evaluate $\left(
\frac{313}{367}
\right)_J$, $\left(
\frac{367}{401}
\right)_J$, $\left(
\frac{401}{313}
\right)_J$.
\medskip
\noindent 2. Show that the congruence $x^6-11x^4+36x^2-36\equiv0\pmod p$ is soluble for every prime $p$. Hint: Factorise $z^3-11z^2+36z-36$.
\medskip
\noindent 3. Suppose that $a\in{\Bbb Z}\backslash\{0\}$, and there is a $b\in{\Bbb Z}$ such that $a=-b^2$. Show that there is an odd positive integer $m$ such that $\left(
\frac{a}{m}
\right)_J=-1$. Deduce that there is an odd prime $p$ such that $\left(
\frac{a}{p}
\right)_J=-1$.
\medskip
\noindent 4. Suppose that $a\in{\Bbb Z}\backslash\{0\}$ and $a=\pm2^ub$ where $u\in{\Bbb N}$ and $b\in{\Bbb N}$ with both $u$ and $b$ odd. Show that there is an odd positive integer $m$ such that $\left(
\frac{a}{m}
\right)_J=-1$. Deduce that there is an odd prime $p$ such that $\left(
\frac{a}{p}
\right)_J=-1$. Hint: Let $m$ be a solution to $m\equiv5\pmod8$, $m\equiv1\pmod b$.
\medskip
\noindent 5. Suppose that $a\in{\Bbb Z}\backslash\{0\}$ and $a=\pm2^{2u}bq^t$ where $u$ is a non-negative integer, $b\in{\Bbb N}$ and $t\in{\Bbb N}$ with both $b$ and $t$ odd, and $q$ is an odd prime. Show that there is an odd positive integer $m$ such that $\left(
\frac{a}{m}
\right)_J=-1$. Deduce that there is an odd prime $p$ such that $\left(
\frac{a}{p}
\right)_J=-1$. Hint: Let $m$ be a solution to $m\equiv1\pmod{4b}$, $m\equiv n\pmod q$ where $n$ is a quadratic non-residue modulo $q$.
\medskip
\noindent {\bf Harder problem}
\medskip
\noindent 6. Show that an integer $a$ is a perfect square if and only if it is a quadratic residue for every prime $p$ not dividing $a$. Questions 3, 4, 5, are relevant. This is a simple example of the ``local-to-global" principle.
\enddocument