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 rE Z mod q by [r]. (1) Prove that Z[√-1] is an integral domain. (2) Find the order of Z[√-1]X, 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) 6: Z[√-1] → Z/qZ, (r+√-1 s) = [r] + [n][s]. is a surjective ring homomorphism.
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 rE Z mod q by [r]. (1) Prove that Z[√-1] is an integral domain. (2) Find the order of Z[√-1]X, 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) 6: Z[√-1] → Z/qZ, (r+√-1 s) = [r] + [n][s]. is a surjective ring homomorphism.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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 E 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+√−1 s) = [r] + [n][s].
is a surjective ring homomorphism.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F008f8cce-1e45-43a4-8b17-46721d7357f5%2F608a04b4-cdef-4184-b5c5-edd83073e87e%2Fbtnti35_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 E 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+√−1 s) = [r] + [n][s].
is a surjective ring homomorphism.
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