5. Let d € Z - {0}, assume that √d & Q, and let σa : Z[√d] → Z[√d] be the conjugation function. (a) Prove that σd(21 +22) = Od(²1) + Od(22) for all 2₁, 22 € Z[√d]. (b) Prove that (²1²2) = 0d(²₁)0d(32) for all 21, 22 € Z[√d]. The formulas established in (a) and (b), along with the fact that σa(1) = 1, amount to the assertion that oa is a ring homomorphism (for those with linear algebra experience: linear transformations are to vector spaces as ring homomorphisms are to rings).
5. Let d € Z - {0}, assume that √d & Q, and let σa : Z[√d] → Z[√d] be the conjugation function. (a) Prove that σd(21 +22) = Od(²1) + Od(22) for all 2₁, 22 € Z[√d]. (b) Prove that (²1²2) = 0d(²₁)0d(32) for all 21, 22 € Z[√d]. The formulas established in (a) and (b), along with the fact that σa(1) = 1, amount to the assertion that oa is a ring homomorphism (for those with linear algebra experience: linear transformations are to vector spaces as ring homomorphisms are to rings).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 61EQ
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![5. Let d € Z - {0}, assume that √d & Q, and let σa : Z[√d] → Z[√d] be the conjugation
function.
(a) Prove that od(21 + 22) = 0d(²1) + Od(22) for all 2₁, 22 € Z[√d].
(b) Prove that od(²1²2) = 0d(²₁)0a(²2) for all 2₁, 22 € Z[√√d].
The formulas established in (a) and (b), along with the fact that σ¿(1) = 1, amount to the
assertion that is a ring homomorphism (for those with linear algebra experience: linear
transformations are to vector spaces as ring homomorphisms are to rings).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2Fb0522b9b-0d98-4a0f-bf1a-3512fdb3ee3e%2Fls6ucu_processed.png&w=3840&q=75)
Transcribed Image Text:5. Let d € Z - {0}, assume that √d & Q, and let σa : Z[√d] → Z[√d] be the conjugation
function.
(a) Prove that od(21 + 22) = 0d(²1) + Od(22) for all 2₁, 22 € Z[√d].
(b) Prove that od(²1²2) = 0d(²₁)0a(²2) for all 2₁, 22 € Z[√√d].
The formulas established in (a) and (b), along with the fact that σ¿(1) = 1, amount to the
assertion that is a ring homomorphism (for those with linear algebra experience: linear
transformations are to vector spaces as ring homomorphisms are to rings).
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