First Supplement to Law of Quadratic Reciprocity/Examples/3

Examples of First Supplement to Law of Quadratic Reciprocity

$-1$ is a quadratic non-residue of $3$.

Proof

From Quadratic Residues of 3:

there exists exactly $1$ quadratic residue modulo $3$, and that is $1$.

We also have that:

$3 = 4 \times 1 - 1$

and so is not of the form $4 n + 1$.

$\blacksquare$

Sources

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