Power residue
modulo
An integer for which the congruence
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is solvable for a given integer . The number
is called a residue of degree
modulo
. If this congruence is not solvable, then
is called a non-residue of degree
modulo
. When
, the power residues and non-residues are said to be quadratic, when
, cubic, and when
, biquadratic.
In the case of a prime modulus , the question of the solvability of the congruence
(
) can be answered by using Euler's test: If
, then for the congruence
(
) to be solvable it is necessary and sufficient that
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When this condition is fulfilled, the original congruence has different solutions modulo
. It follows from this test that among the numbers
there are exactly
residues and
non-residues of degree
modulo
. 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
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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 |
Power residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_residue&oldid=15394