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Difference between revisions of "Cubic residue"

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(→‎Comments: see also Power residue)
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''modulo $m$''
 
''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
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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 the [[Euler criterion]]: The congruence $x^3\equiv a$ ($\bmod\,p$), $(a,p)=1$, is solvable if and only if
  
 
$$a^{(p-1)/q}\equiv1\pmod p,$$
 
$$a^{(p-1)/q}\equiv1\pmod p,$$
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====Comments====
 
====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 $y$.  
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From [[class field theory]] one obtains, e.g., that $2$ is a cubic residue modulo a prime number $p \equiv 1 \pmod 3$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$ (a result conjectured by Euler and first proved by Gauss).
  
 
See also [[Quadratic residue]]; [[Power residue]]; [[Reciprocity laws]]; [[Complete system of residues]]; [[Reduced system of residues]].
 
See also [[Quadratic residue]]; [[Power residue]]; [[Reciprocity laws]]; [[Complete system of residues]]; [[Reduced system of residues]].
  
[[Category:Number theory]]
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====References====
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* Cox, David A. ''Primes of the form $x^2+n y^2$''. John Wiley & Sons (1989) {{ISBN|0-471-50654-0}} {{ZBL|0701.11001}}

Latest revision as of 20:43, 5 December 2023

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

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 the Euler 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 $p \equiv 1 \pmod 3$ if and only if $p$ can be written in the form $p=x^2+27y^2$ with integers $x$ and $y$ (a result conjectured by Euler and first proved by Gauss).

See also Quadratic residue; Power residue; Reciprocity laws; Complete system of residues; Reduced system of residues.

References

  • Cox, David A. Primes of the form $x^2+n y^2$. John Wiley & Sons (1989) ISBN 0-471-50654-0 Zbl 0701.11001
How to Cite This Entry:
Cubic residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cubic_residue&oldid=33705
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article