$() = ^{−1} + ^{−2} + ⋯ + 1$ then $(())^2 = ()$ in $[]/⟨ ^ − 1 ⟩$

Question

If $$ is a finite field with $q$ elements an $n$ . I need to prove that if $() = ^{−1} + ^{−2} + ⋯ + 1$ then $(())^2 = ()$ in $[]/⟨ ^ − 1 ⟩$ I tried to prove it with induction but i am stuck.I would appreciate help thank you

Answer 1

This problem rewrites $$\left(\frac{x^n-1}{x-1}\right)^2\equiv n\frac{x^n-1}{x-1}\bmod{x^n-1}$$ i.e. $$x^n-1\equiv n(x-1)\bmod{(x-1)^2}$$ or equivalently $$(1+y)^n-1\equiv ny\bmod{y^2}.$$ This follows immediatly from the binomial formula.

Note that your finite field can be replaced by any commutative ring.