2. For each of the following pairs (z', 2) of numbers in Z[i]¹, determine whether or not z' divides z in Z[i] by computing z/z' in Q[i]. Show all of your work. (a) (1-2i, 14-3i) (b) (4+5i, 14+3i) (c) (10+ i, 101)
2. For each of the following pairs (z', 2) of numbers in Z[i]¹, determine whether or not z' divides z in Z[i] by computing z/z' in Q[i]. Show all of your work. (a) (1-2i, 14-3i) (b) (4+5i, 14+3i) (c) (10+ i, 101)
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 1EQ: 1. Suppose that, in Example 2.27, 400 units of food A, 600 units of B, and 600 units of C are placed...
Related questions
Question
![If 21, 22 € Z[√d] with 22 ‡ 0, then part (b) of Problem 1 implies that 2₁/22 is an element of
Q[√d]. It is an immediate consequence of the definition of divisibility that zą divides z₁ in
Z[√d] if and only if %₁/22 € Z[√d]. By part (a) of Problem 1, 2₁/22 is an element of Z[√d]
precisely when its rational and irrational parts, which can be computed using the formula
2₂¹ = 0d(22)/Na(22) from part (b), are integers. The upshot is that we can determine whether
or not one nonzero quadratic integer divides another with a straightforward computation in
the field Q[√d].
2. For each of the following pairs (z', z) of numbers in Z[i]¹, determine whether or not z'
divides z in Z[i] by computing z/z' in Q[i]. Show all of your work.
(a) (1 – 2i, 14 — 3i)
(b) (4+5i, 14+ 3i)
(c) (10 + i, 101)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2F9ad049ef-2d27-4dfb-a060-2dff3fc7b059%2Ftg80qb4_processed.jpeg&w=3840&q=75)
Transcribed Image Text:If 21, 22 € Z[√d] with 22 ‡ 0, then part (b) of Problem 1 implies that 2₁/22 is an element of
Q[√d]. It is an immediate consequence of the definition of divisibility that zą divides z₁ in
Z[√d] if and only if %₁/22 € Z[√d]. By part (a) of Problem 1, 2₁/22 is an element of Z[√d]
precisely when its rational and irrational parts, which can be computed using the formula
2₂¹ = 0d(22)/Na(22) from part (b), are integers. The upshot is that we can determine whether
or not one nonzero quadratic integer divides another with a straightforward computation in
the field Q[√d].
2. For each of the following pairs (z', z) of numbers in Z[i]¹, determine whether or not z'
divides z in Z[i] by computing z/z' in Q[i]. Show all of your work.
(a) (1 – 2i, 14 — 3i)
(b) (4+5i, 14+ 3i)
(c) (10 + i, 101)
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