Given the ring of integers < Z, +₁, and the set of equivalence classes Fz = {[k, l] : (k,1) € Zx Z# for all k € Z and 1 e Z#1. Define addition (+) by [k, 1] + [m, n] := [k · n + 1. m, 1-n] and multiplication () by [k, 1] • [m, n] := [k - m, 1. n] for [k, 1], [m, n] € Fz. (a) Determine the unity of Fz, +, >. Justify your answer. (b) Determine the multiplicative inverse of [k, 1] F#. Justify your answer. (c) Show that multiplication is well-defined in the set Fz. (d) Prove that the second distributive law holds in . (e) Show that the subset Kz := {[k, 1] : k € Z with unity 1 € Z} C Fz is a subring of Fz that is isomorphic to Z, i.e. Kz, +, Z, +, >. (f) Prove that the triple is a field of quotients that contains an integral domain K ~ Z; it is also embedded in any field < Rz, +, which contains ZC Rz.
Given the ring of integers < Z, +₁, and the set of equivalence classes Fz = {[k, l] : (k,1) € Zx Z# for all k € Z and 1 e Z#1. Define addition (+) by [k, 1] + [m, n] := [k · n + 1. m, 1-n] and multiplication () by [k, 1] • [m, n] := [k - m, 1. n] for [k, 1], [m, n] € Fz. (a) Determine the unity of Fz, +, >. Justify your answer. (b) Determine the multiplicative inverse of [k, 1] F#. Justify your answer. (c) Show that multiplication is well-defined in the set Fz. (d) Prove that the second distributive law holds in . (e) Show that the subset Kz := {[k, 1] : k € Z with unity 1 € Z} C Fz is a subring of Fz that is isomorphic to Z, i.e. Kz, +, Z, +, >. (f) Prove that the triple is a field of quotients that contains an integral domain K ~ Z; it is also embedded in any field < Rz, +, which contains ZC Rz.
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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Given the ring of integers < Z, +, >, and the set of equivalence classes Fz = {[k, l] : (k,l) €
Zx Z# for all k € Z and 1 € Z#}. Define addition (+) by [k, 1] + [m, n] = [kn+1 m, 1 n] and
multiplication () by [k, 1] • [m, n] := [k · m, 1 - n] for [k, 1], [m, n] = Fz.
(a) Determine the unity of Fz, +, >. Justify your answer.
(b) Determine the multiplicative inverse of [k, 1] F#. Justify your answer.
(c) Show that multiplication is well-defined in the set Fz.
(d) Prove that the second distributive law holds in <Fz, +, . >.
(e) Show that the subset Kz := {[k, 1]: k € Z with unity 1 € Z} C Fz is a subring of Fz that is
isomorphic to Z, i.e. <Kz, +, ~<Z, +, >.
(f) Prove that the triple < Fz, +, > is a field of quotients that contains an integral domain K ~ Z;
it is also embedded in any field < Rz, +, which contains ZC Rz.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb685f800-bd36-484b-82e6-b7cb9e75277f%2Fb0f0af88-c9a8-4876-a76d-9e91183a6f0b%2F30nveqb_processed.png&w=3840&q=75)
Transcribed Image Text:.
Given the ring of integers < Z, +, >, and the set of equivalence classes Fz = {[k, l] : (k,l) €
Zx Z# for all k € Z and 1 € Z#}. Define addition (+) by [k, 1] + [m, n] = [kn+1 m, 1 n] and
multiplication () by [k, 1] • [m, n] := [k · m, 1 - n] for [k, 1], [m, n] = Fz.
(a) Determine the unity of Fz, +, >. Justify your answer.
(b) Determine the multiplicative inverse of [k, 1] F#. Justify your answer.
(c) Show that multiplication is well-defined in the set Fz.
(d) Prove that the second distributive law holds in <Fz, +, . >.
(e) Show that the subset Kz := {[k, 1]: k € Z with unity 1 € Z} C Fz is a subring of Fz that is
isomorphic to Z, i.e. <Kz, +, ~<Z, +, >.
(f) Prove that the triple < Fz, +, > is a field of quotients that contains an integral domain K ~ Z;
it is also embedded in any field < Rz, +, which contains ZC Rz.
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