Question

(a) Show that E ε4 and use it to deduce that α > 0. (b) Show that εσ σ and use it to deduce that ơ--0: (c) Use Part (b) to show that αβ -1 and α + β-_1. (d) Use Part (c) to find a quadratic polynomial in z with real coefficients whose roots (e) Use Part (d) to express cos(2π/5) using radical sign (no trig functions, no deci- are α and β mals).

Answer 1

Here, epsilon = exp(2 pii/5) = e^2pii/5 alpha = epsilon + epsilon^4, beta = epsilon^2 + epsilon^3, sigma = 1 + epsilon + epsilon^2 + epsilon^3 + epsilon^4 clearly, epsilon^5 = e^2pi I = Cos 2 pi + i Sin 2 pi = 1 So, epsilon is a root of x^5 - 1 = 0 so all the roots of the form 1, epsilon, epsilon^2, epsilon^3, epsilon^4, thus all are conjugate except 1 as real root epsilon middot epsilon^4 = epsilon^5 = 1, so epsilon, epsilon^4 and epsilon^2 middot epsilon^3 = epsilon^5 = 1 so epsilon^2, epsilon^3 conjugate clearly epsilon = conjugate of epsilon = epsilon^4 also, epsilon + epsilon64/2 greaterthanorequalto (epsilon middot epsilon^4)^1/2 by AM greaterthanorequalto GM implies alpha/2 greaterthanorequalto 1[because epsilon middot epsilon^4 = epsilon^5 = 1] implies alpha greaterthanorequalto 2 > 0 (proved) Now, epsilon^5 - 1 = 0 implies (epsilon - 1)(1 + epsilon + epsilon^2 + epsilon^3 + epsilon^4) = 0 implies epsilon = 1 & sigma =