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- Id: 736769
- Posted: 2011-12-14 08:54:33
by danbooru - Size: 1296x1812
- Source: img63.pixiv.net/img/laevateinn495/21484864_big_p1.jpg
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That's right. We've seen the primitive nth Roots of Unity here, but there's also the primitive Root of Unity modulo n. Actually, we'll use that to prove something in the next chapter, namely, the Moebius Inversion Formula.
So, the number of primitive nth roots of unity is phi(n).
By definition of the Roots of Unity, the problem is to solve x^6-1=0. The factorization of the equation gives us (x-1)(x+1)(x^2-x+1)(x^2+x+1)=0. The solution to the equation are 1, -1, (1+sqrt(3)i)/2, -(1+sqrt(3)i)/2, (1-sqrt(3)i)/2, -(1-sqrt(3)i)/2. Amongst those, 1 is a 1st root of unity, -1 is a 2nd root of unity, -(1+sqrt(3)i)/2 and -(1-sqrt(3)i)/2 are 3rd roots of unity, and therefore we have 2 primitive 6th roots of unity: (1+sqrt(3)i)/2 and (1-sqrt(3)i)/2.
[Example] Find a primitive 6th Root of Unity.
As is evident from the definition above, if there exists an nth Root of Unity for a k, then it must be that k and n are coprime. For example, if n is a prime number, then all nth Roots of Unity for 0 < k < n will be a primitive nth Root of Unity.
[Definition 12.1] (Primitive nth Root of Unity) If an nth Root of Unity m is never the kth Root of Unity for any natural number k < n, then we call m a primitive nth Root of Unity.
This is an nth Root of Unity. When raised to the nth power, the sine and cosine functions inside will become 2k*pi and become 1.
Let's begin with the introduction of the Roots of Unity. An nth Root of Unity is, as its name suggests, a complex number that equals to 1 when raised to nth power. The Roots of Unity for n is the set of solutions to the equation x^n - 1 = 0.
<a href="http://en.wikipedia.org/wiki/Nth_roots_of_unity">Roots of Unity</a>
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