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The Legendre symbol (a|p) is 1 when a is a non-zero square modulo p, −1 when it is not, and 0 when p divides a. Euler's criterion computes it as a^((p−1)/2) mod p, reading p − 1 as −1.
Euler's criterion
(a|p) ≡ a^((p−1)/2) (mod p)
5⁵ mod 11 = 1, so yes: 4² = 16 ≡ 5 (mod 11).
Exactly (p − 1)/2 non-zero ones, since x and −x always square to the same value.