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Chapter 4

The Binomial Equation

 
 
4.2 The equation  where z and a are complex numbers 

With   it will be  where p>0 the modulus and the argument.
So, if  is one solution of the equation   then:

From the last equation we conclude

and , where k integer.
The equation is true for the complex numbers of the form:
Now, we are going to show that the v values of z that come out from the last equation for k=0,1,2,...,v-1
are all different from each other.
Indeed let's assume that there are  with  such as then the difference of the arguments will be a multiple of .
So, which means that the number  which cannot be true since 0<<v.

So, we found the v different roots of the equation . That means that the equation has exactly v roots.

We have proven the Theorem
 
The equation  has exactly different roots that can be found from the following formula:
where p the modulus and