TheoremΒ A-7.2.1 Formula for -th Root of Unity
Given , there exist exactly different -th roots of unity:
,
for different values, where .
Proof.
TheoremΒ A-7.2.2 Order of the Root of Unity
Given (the complex number domain) and where , is an -th root of unity if and only if .
Proof. We use TheoremΒ A-4.2.1:
TheoremΒ A-7.2.3 Set of All -th Roots of Unity
The set of all -th roots of unity is the union (i.e., the union of all primitive -th roots of unity where ).
Proof.
TheoremΒ A-7.2.4 Condition for Primitive -th Roots of Unity
Given an -th root of unity for where , is a primitive -th root of unity if and only if (i.e., is co-prime to ).
Proof.
TheoremΒ A-7.2.5 The number of Primitive -th Roots of Unity
The number of primitive -th roots of unity is (i.e., the number of elements in that are coprime to ).
Proof.