Cubic residue
modulo
An integer for which the congruence
(
) is solvable. If the congruence has no solution,
is called a cubic non-residue modulo
. If the modulus is a prime number
, the congruence
(
) may be checked for solvability using Euler's criterion: The congruence
(
),
, is solvable if and only if
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where . When the condition is satisfied, the congruence has exactly
distinct solutions modulo
. It follows from the criterion, in particular, that for a prime number
, the sequence of numbers
contains exactly
cubic non-residues and
cubic residues modulo
.
Comments
From class field theory one obtains, e.g., that is a cubic residue modulo a prime number
if and only if
can be written in the form
with integers
and
. See also Quadratic residue; Reciprocity laws; Complete system of residues; Reduced system of residues.
Cubic residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cubic_residue&oldid=13695