PlanetMath: quadratic character of 2

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quadratic character of 2

(Theorem)

For any odd prime , Gauss's lemma quickly yields

$\displaystyle \left( \frac{2}{p} \right)$	$\displaystyle =$	$\displaystyle 1$ if $\displaystyle p\equiv\pm 1\pmod{8}$	(1)
$\displaystyle \left( \frac{2}{p} \right)$	$\displaystyle =$	$\displaystyle -1$ if $\displaystyle p\equiv\pm 3\pmod{8}$	(2)

But there is another way, which goes back to Euler, and is worth seeing, inasmuch as it is the prototype of certain more general arguments about character sums.

Let $\sigma$ be a primitive eighth root of unity in an algebraic closure of $\mathbb{Z}/p\mathbb{Z}$ , and write $\tau=\sigma+\sigma^{-1}$ . We have $\sigma^4=-1$ , whence $\sigma^2+\sigma^{-2}=0$ , whence

$\displaystyle \tau^2=2\;.$

By the binomial formula, we have

$\displaystyle \tau^p=\sigma^p+\sigma^{-p}\;.$

If $p\equiv\pm 1\pmod 8$ , this implies $\tau^p=\tau$ . If $p\equiv\pm 3\pmod 8$ , we get instead $\tau^p=\sigma^5+\sigma^{-5}= -\sigma^{-1}-\sigma=-\tau$ . In both cases, we get $\tau^{p-1}=\left( \frac{2}{p} \right)$ , proving (1) and (2).

A variation of the argument, closer to Euler's, goes as follows. Write

$\displaystyle \sigma=\exp(2\pi i/8)$