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root of unity modulo n
In mathematics, namely ring theory, a ''k''-th root of unity modulo ''n'' for positive integers ''k'', ''n'' ≥ 2, is a solution ''x'' to the equation (or ''congruence'') . If ''k'' is the smallest such exponent for ''x'', then ''x'' is called a primitive ''k''-th root of unity modulo ''n''.〔
〕 See modular arithmetic for notation and terminology.
Do not confuse this with a primitive element modulo ''n'', where the primitive element must generate all units of the residue class ring by exponentiation.
That is, there are always roots and primitive roots of unity modulo ''n'' for ''n'' ≥ 2, but for some ''n'' there is no primitive element modulo ''n''. Being a root or a primitive root modulo ''n'' always depends on the exponent ''k'' and the modulus ''n'', whereas being a primitive element modulo ''n'' only depends on the modulus ''n'' — the exponent is automatically .
== Roots of unity ==
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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