First Supplement to Law of Quadratic Reciprocity/Examples/5

Examples of First Supplement to Law of Quadratic Reciprocity

$-1$ is a quadratic residue of $5$.

Proof

From Quadratic Residues of 5:

the set of quadratic residues modulo $5$ is:

$\set {1, 4}$

We have that:

$4 \equiv -1 \pmod 5$

and so $-1$ is therefore a quadratic residue of $5$.

We also have that:

$5 = 4 \times 1 + 1$

and so is of the form $4 n + 1$.

$\blacksquare$

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Exercise $6$