Primitive Roots of Unity

A field element ω is a primitive root of unity if     for and  

Examples
1.   -1  is a primitive (square) root of unity in the real numbers
2.   In the complex numbers i  is a primitive root of unity
        
3.   3 is a primitive root of unity in the field

4.  11 is a primitive () root of unit in the field of integers modulo  .    

Since p is prime,

This says that  

This factorization tells us that there is an element of order in the group .  It happens that  11 is one of the primitive () roots of unit in .   

Theorem:  In an field F,  the only solutions to the equation   are

Proof:  A solution to is a root of the polynomial  which has the factorization .  This shows that   are root and if is any other element of F,    and are both nonzero, so
        
so   is not a root.  

Theorem:   If is a primitive   root of unity and N is even, then
    (1)    
    (2)   
    (3)    is a primitive root of unity
    (4)