Namespaces
Variants
Actions

Difference between revisions of "Cubic residue"

From Encyclopedia of Mathematics
Jump to: navigation, search
(TeX)
(Category:Number theory)
Line 2: Line 2:
 
''modulo $m$''
 
''modulo $m$''
  
An integer $a$ for which the [[Congruence|congruence]] $x^3=a$ ($\bmod\,m$) is solvable. If the congruence has no solution, $a$ is called a cubic non-residue modulo $m$. If the modulus is a prime number $p$, the congruence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using Euler's criterion: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solvable if and only if
+
An integer $a$ for which the [[congruence]] $x^3=a$ ($\bmod\,m$) is solvable. If the congruence has no solution, $a$ is called a cubic non-residue modulo $m$. If the modulus is a prime number $p$, the congruence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using Euler's criterion: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solvable if and only if
  
 
$$a^{(p-1)/q}\equiv1\pmod p,$$
 
$$a^{(p-1)/q}\equiv1\pmod p,$$
Line 11: Line 11:
  
 
====Comments====
 
====Comments====
From [[Class field theory|class field theory]] one obtains, e.g., that $2$ is a cubic residue modulo a prime number $p$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$. See also [[Quadratic residue|Quadratic residue]]; [[Reciprocity laws|Reciprocity laws]]; [[Complete system of residues|Complete system of residues]]; [[Reduced system of residues|Reduced system of residues]].
+
From [[class field theory]] one obtains, e.g., that $2$ is a cubic residue modulo a prime number $p$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$.  
 +
 
 +
See also [[Quadratic residue]]; [[Reciprocity laws]]; [[Complete system of residues]]; [[Reduced system of residues]].
 +
 
 +
[[Category:Number theory]]

Revision as of 21:09, 16 October 2014

modulo $m$

An integer $a$ for which the congruence $x^3=a$ ($\bmod\,m$) is solvable. If the congruence has no solution, $a$ is called a cubic non-residue modulo $m$. If the modulus is a prime number $p$, the congruence $x^3\equiv a$ ($\bmod\,p$) may be checked for solvability using Euler's criterion: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solvable if and only if

$$a^{(p-1)/q}\equiv1\pmod p,$$

where $q=(3,p-1)$. When the condition is satisfied, the congruence has exactly $q$ distinct solutions modulo $p$. It follows from the criterion, in particular, that for a prime number $p$, the sequence of numbers $1,\dots,p-1$ contains exactly $(q-1)(p-1)/q$ cubic non-residues and $(p-1)/q$ cubic residues modulo $p$.


Comments

From class field theory one obtains, e.g., that $2$ is a cubic residue modulo a prime number $p$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$.

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: http://encyclopediaofmath.org/index.php?title=Cubic_residue&oldid=33704
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article