 Because each "d" is a proper divisor of "n",

so by the above class equation formula_8 must divide "q"−1, and therefore

To see that this forces "n" to be 1, we will show

for "n" > 1 using factorization over the complex numbers.In the polynomial identity

where ζ runs over the primitive "n"-th roots of unity, set "x" to be "q" and then take absolute values

For "n" > 1, we see that for each primitive "n"-th root of unity ζ,

because of the location of "q", 1, and ζ in the complex plane.Thus