please send complete handwritten solution Q3 Problem 3. Consider the quotient ring k defined as follows:k:= Za[X]/(X* + X² +2).(a) Show that k is a finite field and compute its order.(b) What is the order of the multiplicative group k*? By Lagrange's Theorem,what are the possible values for the order of an element of k*?(c) Show that for every integer d 1, there are at most d elements of k oforder d. (Hint: What equation does an element of order d satisfy?)(d) Use the previous parts to show that the group k is cyclic.

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Answered: Consider the quotient ring k defined as…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter5: Rings, Integral Domains, And Fields

Section5.3: The Field Of Quotients Of An Integral Domain

Problem 13E

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