Euler criterion

If an integer $a$ is not divisible by a prime number $p>2$, then the congruence $$ a^{(p-1)/2} \equiv \left({\frac{a}{p}}\right) \pmod p $$ holds, where $\left({\frac{a}{p}}\right)$ is the Legendre symbol. Thus, the Euler criterion gives a necessary and sufficient condition for a number $a \not\equiv 0 \pmod p$ to be a quadratic residue or non-residue modulo $p$. It was proved by L. Euler in 1761 (see [1]).

Euler also obtained a more general result: A number $a \not\equiv 0 \pmod p$ is a residue of degree $n$ modulo a prime number $p$ if and only if $$ a^{(p-1)/\delta} \equiv 1 \pmod p $$ where $\delta = \mathrm{hcf}(p-1,n)$.

Both these assertions carry over easily to the case of a finite field.

References

[1]	L. Euler, "Adnotationum ad calculum integralem Euleri" G. Kowalewski (ed.) , Opera Omnia Ser. 1; opera mat. , 12 , Teubner (1914) pp. 493–538
[2]	I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)

References

[a1]	G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979) pp. Chapts. 5; 7; 8