First Supplement to Law of Quadratic Reciprocity/Examples/11

Examples of First Supplement to Law of Quadratic Reciprocity

$-1$ is a quadratic non-residue of $11$.

Proof

From Quadratic Residues of 11:

the set of quadratic residues modulo $11$ is:

$\set {1, 3, 4, 5, 9}$

but not $10 \equiv -1 \pmod {11}$.

We also have that:

$11 = 4 \times 3 - 1$

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 $6$