Let a complex number `alpha,alpha!=1,` be a rootof hte euation `z^(p+q)-z^p-z^q+1=0,w h e r ep ,q` are distinct primes. Show that either `1+alpha+alpha^2++alpha^(p-1)=0or1+alpha+alpha^2++alpha^(q-1)=0` , but not both together.

Explanation

Given `z^(p+q)-z^p-z^q+1=0`m
`rArr (z^-1)(z^q- 1)=0`
Since `,alpha` is root of Eq. (i) either `alpha^p-1 =0 ` or `alpha^q-1=0`
`rArr " Either " (alpha^p-1)/(alpha -1 )=0 or (alpha ^q-1)/(alpha -1)=0 " " [ as alpha ne 1]`
`rArr " Either " 1+ alpha + alpha ^2+.....+alpha^(q-1)=0`
or ` 1+alpha + ....+alpha^(q-1)=0`
But `alpha^p-1=0` and `alpha^q-1=0` cannot occure simultaneously as p and q are distinct primes so neither P divides q nor q divides p, which is the requirement for `1=alpha^p = alpha^q`