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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/p074630/p07463031.png" /></td> </tr></table>
 
<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/p074630/p07463031.png" /></td> </tr></table>
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463033.png" /> is the Euler function. For a primitive root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463034.png" />, its powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463035.png" /> are incongruent modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463036.png" /> and form a reduced residue system modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463037.png" /> (cf. [[Reduced system of residues|Reduced system of residues]]). Therefore, for each number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463038.png" /> that is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463039.png" /> one can find an exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463040.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463041.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463042.png" />.
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for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463032.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463033.png" /> is the Euler function. For a primitive root <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463034.png" />, its powers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463035.png" /> are incongruent modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463036.png" /> and form a reduced residue system modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463037.png" /> (cf. [[Reduced system of residues|Reduced system of residues]]). Therefore, for each number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463038.png" /> that is relatively prime to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463039.png" /> one can find an exponent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463040.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463041.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463042.png" />: the [[index]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463038.png" /> with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463034.png" />.
  
 
Primitive roots do not exist for all moduli, but only for moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463043.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463045.png" /> is a prime number. In these cases, the multiplicative groups (cf. [[Multiplicative group|Multiplicative group]]) of reduced residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463046.png" /> have the simplest possible structure: they are cyclic groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463047.png" />. The concept of a primitive root modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463048.png" /> is closely related to the concept of the [[Index|index]] of a number modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463049.png" />.
 
Primitive roots do not exist for all moduli, but only for moduli <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463043.png" /> of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463044.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463045.png" /> is a prime number. In these cases, the multiplicative groups (cf. [[Multiplicative group|Multiplicative group]]) of reduced residue classes modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463046.png" /> have the simplest possible structure: they are cyclic groups of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463047.png" />. The concept of a primitive root modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463048.png" /> is closely related to the concept of the [[Index|index]] of a number modulo <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p074/p074630/p07463049.png" />.

Revision as of 22:04, 19 December 2014

A primitive root of unity of order in a field is an element of such that and for any positive integer . The element generates the cyclic group of roots of unity of order .

If in there exists a primitive root of unity of order , then is relatively prime to the characteristic of . An algebraically closed field contains a primitive root of any order that is relatively prime with its characteristic. If is a primitive root of order , then for any that is relatively prime to , the element is also a primitive root. The number of all primitive roots of order is equal to the value of the Euler function if .

In the field of complex numbers, the primitive roots of order take the form

where and is relatively prime to .

A primitive root modulo is an integer such that

for , where is the Euler function. For a primitive root , its powers are incongruent modulo and form a reduced residue system modulo (cf. Reduced system of residues). Therefore, for each number that is relatively prime to one can find an exponent for which : the index of with respect to .

Primitive roots do not exist for all moduli, but only for moduli of the form , where is a prime number. In these cases, the multiplicative groups (cf. Multiplicative group) of reduced residue classes modulo have the simplest possible structure: they are cyclic groups of order . The concept of a primitive root modulo is closely related to the concept of the index of a number modulo .

Primitive roots modulo a prime number were introduced by L. Euler, but the existence of primitive roots modulo an arbitrary prime number was demonstrated by C.F. Gauss (1801).

References

[1] S. Lang, "Algebra" , Addison-Wesley (1984)
[2] C.F. Gauss, "Disquisitiones Arithmeticae" , Yale Univ. Press (1966) (Translated from Latin)
[3] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)


Comments

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Oxford Univ. Press (1979)
How to Cite This Entry:
Primitive root. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Primitive_root&oldid=18612
This article was adapted from an original article by L.V. Kuz'minS.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article