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quadratic polynomial f with rational coefficients u = f(v), v = f(w), w = f(u

Source: Canada Repêchage 2023/7 CMOQR

March 25, 2024
algebrapolynomialtrinomialquadratics

Problem Statement

(a) Let uu, vv, and ww be the real solutions to the equation x3−7x+7=0x^3 - 7x + 7 = 0. Show that there exists a quadratic polynomial ff with rational coefficients such that u=f(v)u = f(v), v=f(w)v = f(w), and w=f(u)w = f(u).
(b) Let uu, vv, and ww be the real solutions to the equation x3−7x+4=0x^3 -7x+4 = 0. Show that there does not exist a quadratic polynomial ff with rational coefficients such that u=f(v)u = f(v), v=f(w)v = f(w), and w=f(u)w = f(u).
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