Welcome guest
You're not logged in.
193 users online, thereof 0 logged in

## Theorem: First Supplementary Law to the Quadratic Reciprocity Law

For a prime number $p > 2$ the following formula for the Legendre symbol holds:

$$\left(\frac {-1}p\right)=(-1)^{\frac{p-1}{2}}.$$

More in detail, this law states that
$$\left(\frac {-1}p\right)=\begin{cases}1&\text{if }p\equiv 1\mod 4,\\-1&\text{if }p\equiv -1\mod 4.\end{cases}$$

In particular, the congruence $x^2(p)\equiv -1(p)$ is only solvable, if $p$ has the form $p\equiv \pm 1\mod 4,$ and any odd prime factor of the integer $x^2+1$ has the form $p\equiv \pm 1\mod 4.$

| | | | | created: 2019-05-26 07:31:42 | modified: 2019-05-26 18:16:07 | by: bookofproofs | references: [1272]

(none)