Power residue

modulo $m$

An integer $a$ for which the congruence $$ x^n \equiv a \pmod m $$ is solvable for a given integer $n > 1$. The number $a$ is called a residue of degree $n$ modulo $m$. If this congruence is not solvable, then $a$ is called a non-residue of degree $n$ modulo $m$. When $n=2$, the power residues and non-residues are said to be quadratic, when $n=3$, cubic, and when $n=4$, biquadratic or quartic.

In the case of a prime modulus $p$, the question of the solvability of the congruence $x^n \equiv a \pmod p$ can be answered by using the Euler criterion: If $q = \mathrm{hcf}(n,p-1)$, then for the congruence $x^n \equiv a \pmod p$ to be solvable it is necessary and sufficient that $$ a^q \equiv 1 \pmod p\ . $$

When this condition is fulfilled, the original congruence has $q$ different solutions modulo $p$. It follows from this test that among the numbers $1,\ldots,p-1$ there are exactly $(p-1)/q$ residues and $(q-1)(p-1)/q$ non-residues of degree $n$ modulo $p$. See Distribution of power residues and non-residues.

Comments

As in the case of quadratic residues one defines a power-residue symbol. Let be a number field containing the -th roots of unity. Let be the ring of integers of and let be a prime ideal of . Let be relatively prime to and . If is a primitive -th root of unity, one has

where is the norm of , i.e. the number of elements of . One now defines the power-residue symbol

If , then is an -th power residue modulo , i.e. () is solvable for . If , and , one finds back the quadratic-residue symbol, cf. Legendre symbol.

There also exist power-residue reciprocity laws, cf. e.g. [a2], which specialize to the quadratic reciprocity law if , .

References

[a1]	W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , Springer & PWN (1990) pp. 394ff
[a2]	J. Neukirch, "Class field theory" , Springer (1986) pp. Chapt. IV, §9