Power residue

2020 Mathematics Subject Classification: Primary: 11A15 [MSN][ZBL]

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 $K$ be a number field containing the $n$-th roots of unity. Let $A$ be the ring of integers of $K$ and let $\mathfrak{p}$ be a prime ideal of $A$. Let $\mathfrak{p}$ be relatively prime to $n$ and $a \in A$. If $\zeta_n$ is a primitive $n$-th root of unity, one has $$ a^{(\mathbf{N}(\mathfrak{p})-1)/n} \equiv \zeta_n^r \pmod {\mathfrak{p}} $$ where $\mathbf{N}(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the number of elements of $A/\mathfrak{p}$. One now defines the power-residue symbol $$ \left({ \frac{a}{\mathfrak{p}} }\right)_n = \zeta_n^r \ . $$

If $\left({ \frac{\alpha}{\mathfrak{p}} }\right)_n = 1$, then $\alpha$ is an $n$-th power residue modulo $\mathfrak{p}$, i.e. $x^n \equiv a \pmod {\mathfrak{p}}$ is solvable for $x \in A$. If $K = \mathbb{Q}$, $n=2$ and $\mathfrak{p} = (p)$, one finds back the quadratic-residue symbol, cf. Legendre symbol.

There also exist power-residue reciprocity laws, cf. e.g. [a2], which specialise to the quadratic reciprocity law if $K = \mathbb{Q}$, $n=2$.

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