Primitive root

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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 .

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)