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

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

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{\bf Reading:  }Ireland and Rosen, Chapter 5 (including the exercises)
and into Chapter~6

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{\bf Problems:}

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1.  (Don't turn this in, but make sure to do it if you haven't yet.)
Confirm for odd primes $p$ that
$$
(-1)^{(p-1)/2}=\begin{cases}
\phantom{-}1&\text{if $p\equiv1\pmod4$,}\\
-1&\text{if $p\equiv3\pmod4$,}
\end{cases}
$$
and
$$
(-1)^{(p^2-1)/8}=\begin{cases}
\phantom{-}1&\text{if $p\equiv\pm1\pmod8$,}\\
-1&\text{if $p\equiv\pm3\pmod8$.}
\end{cases}
$$

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%2.  Use the Gauss Lemma to evaluate the Legendre symbol $(5/p)$ for $p\ne5$ 
%an odd prime, as follows:
%
%(a) Explain why $(5/p)=(-1)^\nu$ where
%$$
%\nu=|\{5, 2\cdot5, 3\cdot5, \dots, (p-1)/2\cdot5\}\cap\{(p/2, p)\cup
%(3p/2,2p)\}|.
%$$
%(The parentheses denote open intervals of real numbers.)
%
%(b) Write $p=20k+r$ where $r\in\{1,3,7,9,11,13,17,19\}$.  Show evaluating 
%$(5/p)$ reduces to finding the parity of
%$$
%\left\{x\in\Z : \frac{r}{10}<x<\frac{2r}{10}\text{ or }
%\frac{3r}{10}<x<\frac{4r}{10}\right\}.
%$$
%Do so for each $r$ to obtain
%$$
%\left(\frac5p\right)=\begin{cases}
%\phantom{-}1&\text{if $p\equiv\pm1,\pm9\pmod{20}$,}\\
%-1&\text{if $p\equiv\pm3,\pm7\pmod{20}$.}
%\end{cases}
%$$
%
%(c) Use Quadratic Reciprocity to show that
%$$
%\left(\frac5p\right)=\begin{cases}
%\phantom{-}1&\text{if $p\equiv1,4\pmod{5}$,}\\
%-1&\text{if $p\equiv2,3\pmod{5}$.}
%\end{cases}
%$$
%Is this consistent with the result in (b)?
%
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2.  Use Quadratic Reciprocity to prove Euler's observation by justifying the 
following computation for distinct odd primes $p$ and $q$ with $4q>p$:
\begin{align*}
\left(\frac q{4q\pm p}\right)&=(-1)^{\frac{q-1}{2}\cdot\frac{4q\pm p-1}{2}}
\left(\frac{4q\pm p}q\right)
=(-1)^{\frac{q-1}{2}\cdot\frac{\pm p-1}{2}}\left(\frac{\pm p}q\right)\\
&=(-1)^{\frac{q-1}{2}\cdot\frac{p-1}{2}}\left(\frac{p}q\right)
=\left(\frac qp\right).
\end{align*}

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3.  Read Ireland and Rosen Exercises 5.3; 5.4--5.8; 5.9, 5.10; 5.11;
5.12, 5.13; 5.14, 5.15; 5.16; 5.17 (just the Proposition); 5.18--5.21;
5.22; 5.23, 5.24; 5.38, and work a small selection of them.

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4.  Work Ireland and Rosen Exercises 5.25--5.28 and/or 5.29--5.31.

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