Cubic residue

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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

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 .


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.

How to Cite This Entry:
Cubic residue. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article