Editing Root of unity (section)

==Elementary properties==
Every {{math|''n''}}th root of unity {{math|''z''}} is a primitive {{math|''a''}}th root of unity for some {{math|''a'' ≤ ''n''}}, which is the smallest positive integer such that {{math|1=''z''<sup>''a''</sup> = 1}}.

Any integer power of an {{math|''n''}}th root of unity is also an {{math|''n''}}th root of unity,<ref name="integer-power">{{cite book
|last = Reilly |first= Norman R.
|year = 2009
|title = Introduction to Applied Algebraic Systems
|url = https://books.google.com/books?id=q33he4hOlKcC&pg=PA137
|page = 137
|publisher= Oxford University Press
|isbn = 978-0-19-536787-4
}}</ref> as
:<math>(z^k)^n = z^{kn} = (z^n)^k = 1^k = 1.</math>
This is also true for negative exponents. In particular, the [[multiplicative inverse|reciprocal]] of an {{math|''n''}}th root of unity is its [[complex conjugate]], and is also an {{math|''n''}}th root of unity:<ref name="conjugate">{{cite book
|last = Rotman|first = Joseph J.|author-link = Joseph J. Rotman
|title = Advanced Modern Algebra
|year = 2015
|edition = 3rd
|volume = 1
|url = https://books.google.com/books?id=SugUCwAAQBAJ&pg=PA129
|page = 129
|publisher = American Mathematical Society
| isbn=9781470415549 }}</ref>
:<math>\frac{1}{z} = z^{-1} = z^{n-1} = \bar z.</math>

If {{math|''z''}} is an {{math|''n''}}th root of unity and {{math|''a'' &equiv; ''b'' (mod ''n'')}} then {{math|1=''z''<sup>''a''</sup> = ''z''<sup>''b''</sup>}}. Indeed, by the definition of [[modular arithmetic|congruence modulo ''n'']], {{math|1=''a'' = ''b'' + ''kn''}} for some integer {{math|''k''}}, and hence
:<math> z^a = z^{b+kn} = z^b z^{kn} = z^b (z^n)^k = z^b 1^k = z^b.</math>

Therefore, given a power {{math|''z''<sup>''a''</sup>}} of {{math|''z''}}, one has {{math|1=''z''<sup>''a''</sup> = ''z''<sup>''r''</sup>}}, where {{math|0 ≤ ''r'' < ''n''}} is the remainder of the [[Euclidean division]] of {{mvar|a}} by {{mvar|n}}.

Let {{math|''z''}} be a primitive {{math|''n''}}th root of unity. Then the powers {{math|''z''}}, {{math|''z''<sup>2</sup>}}, ..., {{math|''z''<sup>''n''−1</sup>}}, {{math|1=''z''<sup>''n''</sup> = ''z''<sup>0</sup> = 1}} are {{math|''n''}}th roots of unity and are all distinct. (If {{math|1=''z''<sup>''a''</sup> = ''z''<sup>''b''</sup>}} where {{math|1 ≤ ''a'' < ''b'' ≤ ''n''}}, then {{math|1=''z''<sup>''b''−''a''</sup> = 1}}, which would imply that {{math|''z''}} would not be primitive.) This implies that {{math|''z''}}, {{math|''z''<sup>2</sup>}}, ..., {{math|''z''<sup>''n''−1</sup>}}, {{math|1=''z''<sup>''n''</sup> = ''z''<sup>0</sup> = 1}} are all of the {{math|''n''}}th roots of unity, since an {{math|''n''}}th-[[degree of a polynomial|degree]] [[polynomial equation]] over a field (in this case the field of complex numbers) has at most {{math|''n''}} solutions.

From the preceding, it follows that, if {{math|''z''}} is a primitive {{math|''n''}}th root of unity, then <math>z^a = z^b</math> [[if and only if]] <math>a\equiv b \pmod{ n}.</math>
If {{math|''z''}} is not primitive then  <math>a\equiv b \pmod{ n}</math> implies <math>z^a = z^b,</math> but the converse may be false, as shown by the following example. If {{math|1=''n'' = 4}}, a non-primitive {{math|''n''}}th root of unity is {{math|1= ''z'' = −1}}, and one has <math>z^2 = z^4 = 1</math>, although <math> 2 \not\equiv 4 \pmod{4}.</math>

Let {{math|''z''}} be a primitive {{math|''n''}}th root of unity. A power {{math|1=''w'' = ''z''<sup>''k''</sup>}} of {{mvar|z}} is a primitive {{math|''a''}}th root of unity for 
:<math> a = \frac{n}{\gcd(k,n)},</math>
where <math>\gcd(k,n)</math> is the [[greatest common divisor]] of {{mvar|n}} and {{mvar|k}}. This results from the fact that {{math|''ka''}} is the smallest multiple of {{mvar|k}} that is also a multiple of {{mvar|n}}. In other words, {{math|''ka''}} is the [[least common multiple]] of {{mvar|k}} and {{mvar|n}}. Thus 
:<math>a =\frac{\operatorname{lcm}(k,n)}{k}=\frac{kn}{k\gcd(k,n)}=\frac{n}{\gcd(k,n)}.</math>

Thus, if {{math|''k''}} and {{math|''n''}} are [[coprime]], {{math|''z<sup>k</sup>''}} is also a primitive {{math|''n''}}th root of unity, and therefore there are {{math|''φ''(''n'')}} distinct primitive {{math|''n''}}th roots of unity (where {{math|''φ''}} is [[Euler's totient function]]). This implies that if {{math|''n''}} is a prime number, all the roots except {{math|+1}} are primitive.

In other words, if {{math|R(''n'')}} is the set of all {{math|''n''}}th roots of unity and {{math|P(''n'')}} is the set of primitive ones, {{math|R(''n'')}} is a [[disjoint union]] of the {{math|P(''n'')}}:

:<math>\operatorname{R}(n) = \bigcup_{d \,|\, n}\operatorname{P}(d),</math>

where the notation means that {{math|''d''}} goes through all the positive [[divisor]]s of {{math|''n''}}, including {{math|1}} and {{math|''n''}}.

Since the [[cardinality]] of {{math|R(''n'')}} is {{math|''n''}}, and that of {{math|P(''n'')}} is {{math|''φ''(''n'')}}, this demonstrates the classical formula

:<math>\sum_{d \,|\, n}\varphi(d) = n.</math>