3.(a) Show that z8 − 1 = (z − 1)(z + 1)(z − i)(z + i)q(z) for some polynomial q(z) which you should determine. (b) [ACS] Show that the following holds: if (is an eighth root of unity but ¢ ‡ 1,−1, i, –i then Çª + AC³ + BC² + CC + D = 0. Here A, B, C, D are certain integers which you should find.

3.(a) Show that z8 − 1 = (z − 1)(z + 1)(z − i)(z + i)q(z) for some polynomial q(z) which you should determine. (b) [ACS] Show that the following holds: if (is an eighth root of unity but ¢ ‡ 1,−1, i, –i then Çª + AC³ + BC² + CC + D = 0. Here A, B, C, D are certain integers which you should find.

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.2: Determinants

Problem 16AEXP

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