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Homomorphisms s.t. f(f(x))=f(x) - OIMU 2006 Problem 7

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August 30, 2010
trigonometrymodular arithmeticsearchcalculusintegrationalgebrafunction

Problem Statement

Consider the multiplicative group A={z∈C∣z2006k=1,0<k∈Z}A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\} of all the roots of unity of degree 2006k2006^k for all positive integers kk.
Find the number of homomorphisms f:A→Af:A\to A that satisfy f(f(x))=f(x)f(f(x))=f(x) for all elements x∈Ax\in A.
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