Problem
ALG-B3-M07-P009 Impossible Degree
#9
★★★☆☆ Level 3 of 5
Prove that there is no \(P\in\mathbb R[x]\) such that \(P(P(x))=x^2+x+1\).
The degree of a composition must be a square of an integer.
A constant \(P\) is impossible. If \(\deg P=d\ge1\), then \(\deg P(P(x))=d^2\). The right-hand side has degree \(2\). We would need \(d^2=2\), impossible for an integer \(d\). Hence no such polynomial exists.
Simple but important degree obstruction.