grandes-ecoles 2018 Q11

grandes-ecoles · France · centrale-maths1__psi Complex numbers 2 Roots of Unity and Cyclotomic Properties

Using the equation (I.1) satisfied by $r_1$ and $r_2$, determine $r_1 r_2$ and $r_1 + r_2$. Deduce that there exists an integer $\ell \in \llbracket 1, n \rrbracket$ and a complex number $\rho$ satisfying $\rho^2 = bc$ such that $$\lambda = a + 2\rho \cos\left(\frac{\ell \pi}{n+1}\right)$$

Using the equation (I.1) satisfied by $r_1$ and $r_2$, determine $r_1 r_2$ and $r_1 + r_2$. Deduce that there exists an integer $\ell \in \llbracket 1, n \rrbracket$ and a complex number $\rho$ satisfying $\rho^2 = bc$ such that
$$\lambda = a + 2\rho \cos\left(\frac{\ell \pi}{n+1}\right)$$

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