(1) The primitive n-th root of unity only equals 1 when it is raised to a power that is a multiple of n.
(2) If ζ is a primitive n-th root of unity, then the n distinct n-th roots of unity are: ζ0, ζ1, ..., ζn-1
Definition 1: Exponent e of an n-th root of unity
The exponent e of an n-th root of unity ζ is the smallest integer such that e is greater than 0 and ζe = 1.
The exponent of 1 is 1.
The exponent of -1 is 2.
Definition 2: Primitive n-th roots of unity
An n-th root of unity is primitive if and only if its exponent is n.
-1 is a primitive square (second) root of unity since (-1)2 = 1.
1 is not a primitive square (second) root of unity since (1)1 = 1.
i, -i are primitive fourth roots of unity.
1, -1 are not primitive fourth roots of unity.
For a root of unity ζ, ζm = 1 if and only if its exponent e divides m.
(1) Assume that e divides m.
(2) Then there exists f such that m = ef.
(3) So ζm = ζef = (ζe)f = (1)f = 1
(4) Assume that ζm = 1
(5) Let d = gcd(e,m)
(6) By Bezout's Identity (see Lemma 1, here), there exists r,s such that:
mr + es = d
ζd = ζmr*ζes = (ζm)r*(ζe)s = (1)r*(1)s = 1
(8) But since e is the smallest number whereby (ζ)e = 1, it follows that e ≤ d.
(9) But since d divides e, it follows that d ≤ e.
(10) It therefore follows (see Lemma 2, here) that d = e.
(11) Since d divides m, it follows that e divides m.
Let ζ, η be roots of unity of exponents e and f.
If e and f are relatively prime, then ζ*η is a root of unity of exponent e*f.
(1) By definition 1 above:
ζe = 1
ηf = 1
(ζ*η)ef = (ζ)ef*(η)ef = (ζe)f*(η
(2) This shows that ζ*η is a root of unity.
(3) Let k be the exponent of (ζ*η).
(4) From Lemma 1 above, we know that k divides e*f.
(ζ*η)k = 1
It follows that:
(ζ*η)k = ζk*ηk = 1
ζk = η-k
(6) Raising both sides to the power of f gives us:
(ζk)f = ζkf = (η-k)f = η(-kf) = (ηf)-k = (1)-k = 1
(7) So by Lemma 1 again, we see that e divides kf.
(8) But gcd(e,f)=-1 so e must divide k. [see Lemma 3, here]
(9) We can make the same argument in step #5 (by switching ζ and η) to show that f must also divide k.
(10) Since gcd(e,f)=1 and e divides k and f divides k, it follows (
(11) Since ef divides k and k divides ef, it follows (
Theorem 3: Powers of a Primitive N-th Root of Unity
Let ζ be a primitive n-th root of unity
every n-th root of unity is a power of ζ
(1) Since ζ is an n-th root of unity, all powers of ζ are n-th roots of unity:
(ζi)n = (ζ)in = (ζn)i = (1)i = 1
(2) Next, we will show that each of these n powers (ζ0, ζ1, ..., ζn-1) are distinct.
(3) Assume that they are not distinct.
(4) Then, there exists a distinct i,j such that:
0 ≤ i,j ≤ n-1
i ≠ j
ζi = ζj
(5) Assume that j is greater than i.
(6) From step #4, we have:
1 = ζj/ζi = ζj-i
(7) But since j,i are less than n, it follows that j-i is greater than 0 and less than n.
(8) But this is impossible since by definition, we know that n is the exponent of ζ
(9) So, we reject our assumption in step #5
(10) We know that there are exactly n roots of unity by applying the Fundamental Theorem of Algebra which says that xn - 1 = 0 has exactly n roots. [See Theorem, here]
- Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001