A-7.2 Theorems

⟨Theorem A-7.2.1⟩ Formula for 𝐧-th Root of Unity

Given ΞΆn = 1, there exist exactly n different n-th roots of unity:

ΞΆ = e2π‘˜πœ‹π‘–βˆ•n = cos⁑ (2π‘˜πœ‹ n ) + i β‹…sin⁑ (2π‘˜πœ‹ n ),

for n different k values, where k = {0,1,β‹―,n βˆ’1}.

Proof.

1.
Suppose ΞΆ = e2π‘˜πœ‹π‘–βˆ•n. Then, ΞΆn = (e2π‘˜πœ‹π‘–βˆ•n)n = e2π‘˜πœ‹π‘–, and since ΞΆn = 1, we need to find the k values such that e2π‘˜πœ‹π‘– = 1
2.
Euler’s formula states that eiβ‹…x = cos(x) + i β‹…sin(x). Therefore, if x = 2π‘˜πœ‹, then e2π‘˜πœ‹π‘– = cos(2π‘˜πœ‹) + i β‹…sin(2π‘˜πœ‹). This formula becomes 1 if k = 0,1,2,.... Thus, e2π‘˜πœ‹π‘– = 1 for any integer k β‰₯0.
3.
If ΞΆ = e2π‘˜πœ‹π‘–βˆ•n = cos(2π‘˜πœ‹ n ) + i β‹…sin(2π‘˜πœ‹ n ), then the first n roots for k = 0,1,...Β n βˆ’1 are all distinct values, because they lie on the circle in the complex plane (where x-axis is a real value and y-axis is a complex value coefficient) at each angle 2π‘˜πœ‹βˆ•n for k = {0,1,β‹―,n βˆ’1}.
4.
Note ΞΆn = 1 is an n-th polynomial, so it can have at most n roots. Thus, we can consider the first n roots e2π‘˜πœ‹π‘–βˆ•n for k = {0,1,β‹―,n βˆ’1} as the n distinct roots and ignore the rest of roots (i.e., k β‰₯n), considering them to be repetitions of the first n roots on a circle in the complex plane (see FigureΒ 3).
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PIC

FigureΒ 3: The figure illustrates a circle of Euler’s formula in the complex plane (Source)

⟨Theorem A-7.2.2⟩ Order of the Root of Unity

Given ΞΆ ∈ β„‚ (the complex number domain) and ΞΆn = 1 where n β‰₯1, ΞΆ is an n-th root of unity if and only if π—ˆπ—‹π–½β„‚(ΞΆ)Β |Β n.

Proof. We use TheoremΒ A-4.2.1:

1.
Forward Proof: Since π—ˆπ—‹π–½β„‚(ΞΆ) = k is the smallest integer such that ΞΆk = 1, for any n that satisfies ΞΆn = 1, n must be a multiple of k. This means that kΒ |Β n.
2.
Backward Proof: If kΒ |Β n, then n is a multiple of k, which means that ΞΆn = 1.
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⟨Theorem A-7.2.3⟩ Set of All 𝐧-th Roots of Unity

The set of all n-th roots of unity is the union ⋃ ⁑ d|nP(d) (i.e., the union of all primitive d-th roots of unity where dΒ |Β n).

Proof.

1.
Let Ο‰ = e2πœ‹π‘–βˆ•n. Given ΞΆn = 1, for each n-th root of unity ΞΆ is, ΞΆ = Ο‰ki for ki = {0,1,β‹―,n βˆ’1}. Note that according to TheoremΒ A-4.2.1, (Ο‰ki)n = 1 if and only if π—ˆπ—‹π–½β„‚(Ο‰ki)Β |Β n.
2.
Let π—ˆπ—‹π–½β„‚(Ο‰ki) = di. Then, (Ο‰ki)di = 1. Combining these two facts, each n-th root of unity Ο‰ki is also a primary di-th root of unity (i.e., a solution for ΞΆdi = 1), that is, Ο‰ki ∈P(di).
3.
Remember that for each π—ˆπ—‹π–½π”½(Ο‰ki) = di, diΒ |Β n. For every di that divides n, all the (primary) di-th roots of unity are also the n-th root of unity. This is because the (primary) di-th root of unity that satisfies ΞΆdi = 1 also satisfies ΞΆn = 1 (as n is a multiple of di).
4.
Step 2 concludes that each n-th root of unity is a primitive di-th root of unity for some di that divides n. Step 3 concludes that each di-th root of unity, where di divides n, is also the n-th root of unity. Combining these two conclusions, the set of all primitive n-th root of unity is equivalent to the union of all primary di-th roots of unity where di divides n (i.e., ⋃ ⁑ d|nP(d)).
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⟨Theorem A-7.2.4⟩ Condition for Primitive 𝐧-th Roots of Unity

Given an n-th root of unity ΞΆ = Ο‰k for k = {0,1,β‹―,n βˆ’1} where Ο‰ = e2πœ‹π‘–βˆ•n, ΞΆ is a primitive n-th root of unity if and only if gcd(n,k) = 1 (i.e., k is co-prime to n).

Proof.

1.
Note that ΞΆn = 1 and ΞΆ = Ο‰ for k = 1. Thus, π—ˆπ—‹π–½β„‚(Ο‰) = n.
2.
TheoremΒ A-4.2.2 states that if π‘œπ‘Ÿd𝔽(a) = k, then for any n β‰₯1, π‘œπ‘Ÿd𝔽(an) = k gcd(k,n). Similarly, if π—ˆπ—‹π–½β„‚(Ο‰) = n, then for any k β‰₯1, π—ˆπ—‹π–½β„‚(Ο‰k) = n gcd(k,n).
3.
Step 2 implies that π—ˆπ—‹π–½β„‚(Ο‰k) = n (i.e., Ο‰k is a primitive n-th root of unity) if and only if gcd(k,n) = 1.
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⟨Theorem A-7.2.5⟩ The number of Primitive 𝐧-th Roots of Unity

The number of primitive n-th roots of unity is Ο•(n) (i.e., the number of elements in {1,β‹―,n βˆ’1} that are coprime to n).

Proof.

1.
Given ΞΆn = 1, the roots of unity are ΞΆ = Ο‰k where Ο‰ = e2πœ‹π‘–βˆ•n and k = {0,1,β‹―,n βˆ’1}
2.
By definition, Ο‰k is a primitive n-th root of unity if and only if π—ˆπ—‹π–½β„‚(Ο‰k) = n.
3.
Ο‰ is a primitive n-th root of unity, because π—ˆπ—‹π–½β„‚(Ο‰) = n.
4.
According to TheoremΒ A-4.2.2, if π—ˆπ—‹π–½β„‚(Ο‰) = n, then π—ˆπ—‹π–½β„‚(Ο‰k) = n gcd(k,n). Therefore, in order for π—ˆπ—‹π–½β„‚(Ο‰k) = n, gcd(k,n) has to be 1. In other words, k and n have to be co-prime. The total number of such co-primes between n and k = {1,2,β‹―,n βˆ’1} (excluding 0 because gcd(0,n) = n and also π—ˆπ—‹π–½β„‚(Ο‰0) = π—ˆπ—‹π–½β„‚(1) = 1β‰ n) is Ο•(n), which corresponds to the total number of the primitive n-the root of unity.
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