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 $p\equiv \pm 1\mod 4.$

Proofs: 1

Proofs: 1
Solutions: 2

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  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927