Consider the quotient ring k defined as follows: k:= Za[X]/(X* +X² +2). (a) Show thatk 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 of order d. (Hint: What equation does an element of order d satisfy?) (d) Use the previous parts to show that the group k is cyclic.
Consider the quotient ring k defined as follows: k:= Za[X]/(X* +X² +2). (a) Show thatk 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 of order d. (Hint: What equation does an element of order d satisfy?) (d) Use the previous parts to show that the group k is cyclic.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![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 of
order d. (Hint: What equation does an element of order d satisfy?)
(d) Use the previous parts to show that the group k is cyclic.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c7a5306-15ff-4c28-af31-3b9d67d52d5e%2F49078cca-464d-4b58-815e-cdd05583862a%2Fozb0rhg_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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 of
order 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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