grandes-ecoles 2014 QIII.B.2

grandes-ecoles · France · centrale-maths2__mp Roots of polynomials Divisibility and minimal polynomial arguments

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with the polynomial $P$ under composition. Every non-constant polynomial commuting with $P_\alpha$ is monic.
Deduce that, for every integer $n \geqslant 1$, there exists at most one polynomial of degree $n$ that commutes with $P_\alpha$. Determine $\mathcal{C}(X^2)$.

For $\alpha \in \mathbb{C}$, we set $P_\alpha = X^2 + \alpha$. We denote by $\mathcal{C}(P)$ the set of complex polynomials that commute with the polynomial $P$ under composition. Every non-constant polynomial commuting with $P_\alpha$ is monic.

Deduce that, for every integer $n \geqslant 1$, there exists at most one polynomial of degree $n$ that commutes with $P_\alpha$. Determine $\mathcal{C}(X^2)$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.B.1 QI.B.2 QII.A.1 QII.A.2 QII.B.1 QII.B.2 QIII.A.1 QIII.A.2 QIII.B.1 QIII.B.2 QIII.B.3 QIII.B.4 QIII.C.1 QIII.C.2 QIV.A QIV.B QIV.C.1 QIV.C.2 QIV.C.3 QIV.C.4