Proving relations amongst the ninth roots of unity, proving the validity of the symmetric functions for  

 

In order to prove the validity of the symmetric functions for , the minimal polynomial for the primitive ninth roots of unity, it’s necessary to prove certain additive and multiplicative relations amongst the ninth roots of unity, which are,

 

 

At present, these are the algebraic and numeric representations I have for the ninth roots of unity:

 

 

These are the basic relations I need to prove in order to prove the validity of the symmetric relations between roots and coefficients.  I will prove them with two graphs and a short written explanation.

 

 

 

Additive Relations

 

 

 

Recall the ratios of the sides of a 30-60-90 right triangle…

 

 

Multiplicative Relations

 

 

As some of you are probably aware, Geometer’s SketchPad doesn’t seem to be aware of the convention of labeling clockwise angles negative.  Angle ABD should be negative.

 

 

 

There’s one more multiplicative relation I’ll need to prove.

 

 

 

Now, with these tools, it’s easy to prove the validity of the symmetric relations between the roots and coefficients for :