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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