We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $a$ and $\lambda$ be non-zero complex numbers. Assume that $\frac { a } { \lambda } \notin \mathcal { V } _ { p }$, which means that either $a = \lambda$ or $\frac { a ^ { p } } { \lambda ^ { p } } \neq 1$. Prove that the complex number $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } a ^ { j }$ is non-zero.
We fix a natural number $p$ greater than or equal to 2. We denote $\mathcal { V } _ { p } = \left\{ \mathrm { e } ^ { \frac { 2 \mathrm { i } k \pi } { p } } ; k \in \llbracket 1 , p - 1 \rrbracket \right\}$, the set of $p$-th roots of unity different from 1. Let $a$ and $\lambda$ be non-zero complex numbers. Assume that $\frac { a } { \lambda } \notin \mathcal { V } _ { p }$, which means that either $a = \lambda$ or $\frac { a ^ { p } } { \lambda ^ { p } } \neq 1$. Prove that the complex number $\sum _ { j = 0 } ^ { p - 1 } \lambda ^ { p - 1 - j } a ^ { j }$ is non-zero.