(4) Determine ker ø. (5) Prove that the quotient ring Z[√-1] In is a field if and only if q = n² + 1 is prime.
(4) Determine ker ø. (5) Prove that the quotient ring Z[√-1] In is a field if and only if q = n² + 1 is prime.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
How do you do question 4 and 5?
![The following assignment will use repeatedly the following ring
Z[√ 1] = {r + √-1s : r, s € Z},
the Gaussian integers. This is a subset of C and it is a commutative ring. (You don't need to prove these properties -
you can assume them.)
-
Throughout this question let 0 ‡ n € Z and let In = z[√=1]{√−1− n). Set q = n² + 1 and denote the equivalence
class of r € Z mod q by [r].
(1) Prove that Z[√-1] is an integral domain.
(2) Find the order of Z[√-1]×, the group of units of Z[√-1], and determine whether or not it is cyclic.
(3) Show that the map (for all r, s € Z)
: Z[√1] → Z/qZ, (r+√√-1s) = [r] + [n][s].
is a surjective ring homomorphism.
(4) Determine ker ø.
(5) Prove that the quotient ring
Z[√-1
In
is a field if and only if q = n² + 1 is prime.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F596cc392-f3f2-4108-92f0-09890526f49b%2F3856213f-88f7-4dcf-bba0-73b69ae01a5f%2Fxo0fuea_processed.png&w=3840&q=75)
Transcribed Image Text:The following assignment will use repeatedly the following ring
Z[√ 1] = {r + √-1s : r, s € Z},
the Gaussian integers. This is a subset of C and it is a commutative ring. (You don't need to prove these properties -
you can assume them.)
-
Throughout this question let 0 ‡ n € Z and let In = z[√=1]{√−1− n). Set q = n² + 1 and denote the equivalence
class of r € Z mod q by [r].
(1) Prove that Z[√-1] is an integral domain.
(2) Find the order of Z[√-1]×, the group of units of Z[√-1], and determine whether or not it is cyclic.
(3) Show that the map (for all r, s € Z)
: Z[√1] → Z/qZ, (r+√√-1s) = [r] + [n][s].
is a surjective ring homomorphism.
(4) Determine ker ø.
(5) Prove that the quotient ring
Z[√-1
In
is a field if and only if q = n² + 1 is prime.
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