Suppose A € M₂(F) has multiplicative order 5. First, show that x³ – 1 = (x² − 4x + 1)(x² +5x + 1) in F19 [x], where F19 is the field with 19 elements, i.e. Z19. Use this fact to determineall similarity classes of elements of M₂(F19) of multiplicative order 5.

Given: A∈M2(F) has multiplicative order 5.

Answered: Suppose A € M₂(F) has multiplicative…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Suppose A € M₂(F) has multiplicative order 5. First, show that x³ – 1 = (x² − 4x + 1)(x² + 5x + 1) in F19 [x], where F19 is the field with 19 elements, i.e. Z₁9. Use this fact to determine all similarity classes of elements of M₂(F19) of multiplicative order 5.