(b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in Z/61Z by (a): they are +27+ 61Z (you do not need to verify this). Using this information, give the roots of the polynomial X² – X + 19 in Z/61Z.

(b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in Z/61Z by (a): they are +27+ 61Z (you do not need to verify this). Using this information, give the roots of the polynomial X² – X + 19 in Z/61Z.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 33E: Use the fact that 3 is a prime to prove that there do not exist nonzero integers a and b such that...

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(b) Since 61 is a prime with 61 = 1 (mod 12) the polynomial X? + 3 has roots in
Z/61Z by (a): they are +27+ 61Z (you do not need to verify this). Using this
information, give the roots of the polynomial X² – X + 19 in Z/61Z.

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