MCS 494 solution to exercise 4.1

Exercise 4.1: Verify Gauss relation:

     n-1
     ---                  a
     \     ak   {  n  if w  = 1
     /    w   = {
     ---        {  0  otherwise
     k=0

Let z be the sum at the left of the equation above and we derive

           n-1              n          n-1
           ---             ---         ---
     a     \     (a+1)k    \     ak    \     ak    na
    w  z = /    w        = /    w    = /    w   + w
           ---             ---         ---
	   k=0             k=1         k=1

Since w is a root of unity, we have w^n = 1, and thus also w^(na) = 1 = w^(0a). So the above is equivalent to

           n-1               n-1 
           ---               ---  
     a     \     ak     a0   \     ak
    w  z = /    w    + w   = /    w   = z
           ---               ---     
	   k=0               k=0

From the equation (w^a-1)z = 0, we have two possibilities. Either w^a = 1, and then z = n. Or z = 0.