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I have two functions, $p_1,p_2\in\mathcal{H}(\mathbb{C})$ verifying $p_1(p_2(z))=z^2$. I have to prove that $p_1$ and $p_2$ are polynomials and, in addition, show that one of them has degree equal to 1 and the other equal to 2.

After reading this question I know how to solve the first part, but I have no idea how to prove the second one.

Antonio Gamiz Delgado
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1 Answers1

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If $p_1$ and $p_2$ are polynomials, then $\deg(p_1 \circ p_2) = \deg (p_1) \cdot \deg (p_2)$. But $\deg(p_1 \circ p_2) = \deg(z^2) = 2$, and this only factorises as $1 \cdot 2$ or $2\cdot 1$.

hdighfan
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