True/False. For each of the following statements, determine whether they are true or false. If true, you must provide a short proof of that statement. If false, you must provide a counterexample and justify why it is a counterexample to that statement. No marks will be given for unjustified "true" or "false" answers. (a) The element 6+i is irreducible in the ring Z[i] of Gaussian integers. (b) The element 17 is irreducible in the ring Z[i] of Gaussian integers. (c) There is no element of order 4 in the symmetric group S3. (d) There is no element of order 4 in the alternating group As.
True/False. For each of the following statements, determine whether they are true or false. If true, you must provide a short proof of that statement. If false, you must provide a counterexample and justify why it is a counterexample to that statement. No marks will be given for unjustified "true" or "false" answers. (a) The element 6+i is irreducible in the ring Z[i] of Gaussian integers. (b) The element 17 is irreducible in the ring Z[i] of Gaussian integers. (c) There is no element of order 4 in the symmetric group S3. (d) There is no element of order 4 in the alternating group As.
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 1. True/False. For each of the following statements, determine whether they are
true or false. If true, you must provide a short proof of that statement. If false,
you must provide a counterexample and justify why it is a counterexample to that
statement. No marks will be given for unjustified "true" or "false" answers.
(a) The element 6+ i is irreducible in the ring Z[i] of Gaussian integers.
(b) The element 17 is irreducible in the ring Z[i] of Gaussian integers.
(c) There is no element of order 4 in the symmetric group S3.
(d) There is no element of order 4 in the alternating group A5.
Problem 2. Consider the multiplicative group Z, of invertible elements of the ring Z16.
(a) Give a list of all the elements of Z, (justify your reasoning). Each of them
must be written in the form k for some k E (0,..., 15}.
(b) Show that for every a E Z, we have a? = I
(c) Use part (b) to deduce that a = ī for every a E Z. Is Z a cyclic group?
or](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4c7a5306-15ff-4c28-af31-3b9d67d52d5e%2F48f225cb-8d11-45f4-9324-3d0ceb057216%2Fsyk1f56_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1. True/False. For each of the following statements, determine whether they are
true or false. If true, you must provide a short proof of that statement. If false,
you must provide a counterexample and justify why it is a counterexample to that
statement. No marks will be given for unjustified "true" or "false" answers.
(a) The element 6+ i is irreducible in the ring Z[i] of Gaussian integers.
(b) The element 17 is irreducible in the ring Z[i] of Gaussian integers.
(c) There is no element of order 4 in the symmetric group S3.
(d) There is no element of order 4 in the alternating group A5.
Problem 2. Consider the multiplicative group Z, of invertible elements of the ring Z16.
(a) Give a list of all the elements of Z, (justify your reasoning). Each of them
must be written in the form k for some k E (0,..., 15}.
(b) Show that for every a E Z, we have a? = I
(c) Use part (b) to deduce that a = ī for every a E Z. Is Z a cyclic group?
or
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