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''modulo $m$''
 
''modulo $m$''
  
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====Comments====
 
====Comments====
As in the case of quadratic residues one defines a power-residue symbol. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423033.png" /> be a number field containing the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423034.png" />-th roots of unity. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423035.png" /> be the ring of integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423036.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423037.png" /> be a prime ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423038.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423039.png" /> be relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423041.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423042.png" /> is a primitive <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423043.png" />-th root of unity, one has
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As in the case of quadratic residues one defines a power-residue symbol. Let $K$ be a number field containing the $n$-th roots of unity. Let $A$ be the ring of integers of $K$ and let $\mathfrak{p}$ be a prime ideal of $A$. Let $\mathfrak{p}$ be relatively prime to $n$ and $a \in A$. If $\zeta_n$ is a primitive $n$-th root of unity, one has
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423044.png" /></td> </tr></table>
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a^{(\mathbf{N}(\mathfrak{p})-1)/n} \equiv \zeta_n^r \pmod {\mathfrak{p}}
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423045.png" /> is the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423046.png" />, i.e. the number of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423047.png" />. One now defines the power-residue symbol
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where $\mathbf{N}(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the number of elements of $A/\mathfrak{p}$. One now defines the power-residue symbol
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423048.png" /></td> </tr></table>
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\left({ \frac{a}{\mathfrak{p}} }\right)_n = \zeta_n^r \ .
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$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423049.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423050.png" /> is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423051.png" />-th power residue modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423052.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423054.png" />) is solvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423055.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423057.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423058.png" />, one finds back the quadratic-residue symbol, cf. [[Legendre symbol|Legendre symbol]].
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If $\left({ \frac{\alpha}{\mathfrak{p}} }\right)_n = 1$, then $\alpha$ is an $n$-th power residue modulo $\mathfrak{p}$, i.e. $x^n \equiv a \pmod {\mathfrak{p}}$ is solvable for $x \in A$. If $K = \mathbb{Q}$, $n=2$ and $\mathfrak{p} = (p)$, one finds back the quadratic-residue symbol, cf. [[Legendre symbol]].
  
There also exist power-residue reciprocity laws, cf. e.g. [[#References|[a2]]], which specialize to the quadratic reciprocity law if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074230/p07423060.png" />.
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There also exist power-residue reciprocity laws, cf. e.g. [[#References|[a2]]], which specialise to the [[quadratic reciprocity law]] if $K = \mathbb{Q}$, $n=2$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Narkiewicz,  "Elementary and analytic theory of algebraic numbers" , Springer &amp; PWN  (1990)  pp. 394ff</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neukirch,  "Class field theory" , Springer  (1986)  pp. Chapt. IV, §9</TD></TR></table>
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<table>
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<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Narkiewicz,  "Elementary and analytic theory of algebraic numbers" , Springer &amp; PWN  (1990)  pp. 394ff</TD></TR>
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<TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Neukirch,  "Class field theory" , Springer  (1986)  pp. Chapt. IV, §9</TD></TR>
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</table>

Latest revision as of 19:32, 19 December 2014

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

modulo $m$

An integer $a$ for which the congruence $$ x^n \equiv a \pmod m $$ is solvable for a given integer $n > 1$. The number $a$ is called a residue of degree $n$ modulo $m$. If this congruence is not solvable, then $a$ is called a non-residue of degree $n$ modulo $m$. When $n=2$, the power residues and non-residues are said to be quadratic, when $n=3$, cubic, and when $n=4$, biquadratic or quartic.

In the case of a prime modulus $p$, the question of the solvability of the congruence $x^n \equiv a \pmod p$ can be answered by using the Euler criterion: If $q = \mathrm{hcf}(n,p-1)$, then for the congruence $x^n \equiv a \pmod p$ to be solvable it is necessary and sufficient that $$ a^q \equiv 1 \pmod p\ . $$

When this condition is fulfilled, the original congruence has $q$ different solutions modulo $p$. It follows from this test that among the numbers $1,\ldots,p-1$ there are exactly $(p-1)/q$ residues and $(q-1)(p-1)/q$ non-residues of degree $n$ modulo $p$. See Distribution of power residues and non-residues.


Comments

As in the case of quadratic residues one defines a power-residue symbol. Let $K$ be a number field containing the $n$-th roots of unity. Let $A$ be the ring of integers of $K$ and let $\mathfrak{p}$ be a prime ideal of $A$. Let $\mathfrak{p}$ be relatively prime to $n$ and $a \in A$. If $\zeta_n$ is a primitive $n$-th root of unity, one has $$ a^{(\mathbf{N}(\mathfrak{p})-1)/n} \equiv \zeta_n^r \pmod {\mathfrak{p}} $$ where $\mathbf{N}(\mathfrak{p})$ is the norm of $\mathfrak{p}$, i.e. the number of elements of $A/\mathfrak{p}$. One now defines the power-residue symbol $$ \left({ \frac{a}{\mathfrak{p}} }\right)_n = \zeta_n^r \ . $$

If $\left({ \frac{\alpha}{\mathfrak{p}} }\right)_n = 1$, then $\alpha$ is an $n$-th power residue modulo $\mathfrak{p}$, i.e. $x^n \equiv a \pmod {\mathfrak{p}}$ is solvable for $x \in A$. If $K = \mathbb{Q}$, $n=2$ and $\mathfrak{p} = (p)$, one finds back the quadratic-residue symbol, cf. Legendre symbol.

There also exist power-residue reciprocity laws, cf. e.g. [a2], which specialise to the quadratic reciprocity law if $K = \mathbb{Q}$, $n=2$.

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
How to Cite This Entry:
Power residue. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Power_residue&oldid=35694
This article was adapted from an original article by S.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article