The subset 2 of C is closed under the addition and multiplication operations on C; it is also closed under subtraction, and contains 0 and 1.¹ These properties make Z an example of a subring of C. A large class of subrings of C can be constructed as follows. For every integer d € Z - {0}, the polynomial X²-d has exactly two roots in C (by the fundamental theorem of algebra!): if we denote one of the roots by √d, then the other is -√d. Define the subset Z[√d] of C by Z[√d] = {z € C: z = a +b√d for some a, b = Z}. The notation Z[√d] is read "Z adjoin √d." We use this terminology because (it turns out) Z[√d] is the smallest subring of C that contains Z and √d. By definition, the elements of Z[√d] are the complex numbers having a particular form: a +b√d with a, b € Z. This means that, to prove that a complex number is an element of Z[√d], we must show that the number can be expressed in the indicated form. Keep this in mind when solving the first problem below. 1. Let de Z-{ (a) Using the definition of Z[√d], explain why Z is a subset of Z[√d]. (b) Prove that Z[√d] is closed under the addition operation on C. (c) Prove that Z[√d] is closed under the multiplication operation on C.
The subset 2 of C is closed under the addition and multiplication operations on C; it is also closed under subtraction, and contains 0 and 1.¹ These properties make Z an example of a subring of C. A large class of subrings of C can be constructed as follows. For every integer d € Z - {0}, the polynomial X²-d has exactly two roots in C (by the fundamental theorem of algebra!): if we denote one of the roots by √d, then the other is -√d. Define the subset Z[√d] of C by Z[√d] = {z € C: z = a +b√d for some a, b = Z}. The notation Z[√d] is read "Z adjoin √d." We use this terminology because (it turns out) Z[√d] is the smallest subring of C that contains Z and √d. By definition, the elements of Z[√d] are the complex numbers having a particular form: a +b√d with a, b € Z. This means that, to prove that a complex number is an element of Z[√d], we must show that the number can be expressed in the indicated form. Keep this in mind when solving the first problem below. 1. Let de Z-{ (a) Using the definition of Z[√d], explain why Z is a subset of Z[√d]. (b) Prove that Z[√d] is closed under the addition operation on C. (c) Prove that Z[√d] is closed under the multiplication operation on C.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Problem 1
![The subset Z of C is closed under the addition and multiplication operations on C; it is
also closed under subtraction, and contains 0 and 1. These properties make Z an example
of a subring of C. A large class of subrings of C can be constructed as follows. For every
integer d € Z - {0}, the polynomial X²-d has exactly two roots in C (by the fundamental
theorem of algebra!): if we denote one of the roots by √d, then the other is -√d. Define
the subset Z[√d] of C by
Z[√d] = {z € C: z = a +b√d for some a, b = Z}.
The notation Z[√d] is read "Z adjoin √d." We use this terminology because (it turns out)
Z[√d is the smallest subring of C that contains Z and √d. By definition, the elements
of Z[√d] are the complex numbers having a particular form: a+b√d with a, b € Z. This
means that, to prove that a complex number is an element of Z[√d], we must show that the
number can be expressed in the indicated form. Keep this in mind when solving the first
problem below.
1. Let de Z-{0}.
(a) Using the definition of Z[√d], explain why Z is a subset of Z[√d].
(b) Prove that Z[√d] is closed under the addition operation on C.
(c) Prove that Z[√d] is closed under the multiplication operation on C.
(You will not actually need the assumption that d 0 in your arguments; but if d = 0,
then Z[√d] = Z, and we are most interested in the case where √d & Q, as this ensures that
Z[√d properly contains Z.)
2. Let d € Z - {0} and assume that √d & Q (we will eventually prove that this is true if
and only if d is not the square of an integer, i.e., if and only if √d & Z). In this case, the
elements of Z[√d are called quadratic integers.
(a) Prove that, if z € Z[√d], then there are unique integers a and b such that z = a+b√d.
(Hint: Such integers exist by the definition of Z[√d]. Assuming that there are integers
a₁, a2, b₁, and by such that z = a₁ + b₁ √d and z= a₂ + b₂√d, we need to show that
0₁ 0₂ and b₁ b₂. Ignoring the first equality, suppose for the sake of a contradiction
that b₁ b₂. Use the irrationality of √d to reach a contradiction. How does the
desired result follow?)
(b) We would like to define a function
=
: Z[√d] → Z[√d] by
a(z) = a - b√d,
where a, b € Z satisfy 2 = a +b√d. Why is there a potential issue with trying to
define a function using this formula? Explain how (a) resolves this issue, so that the
displayed formula does indeed give a well-defined function, which we call conjugation.
(Hint: compare with the first problem of Homework 1.)
(c) Prove that the function o defined in (b) is bijective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2F00481e6c-fc35-43e5-a7f1-55c2f4f9a880%2Fapm6a3j_processed.png&w=3840&q=75)
Transcribed Image Text:The subset Z of C is closed under the addition and multiplication operations on C; it is
also closed under subtraction, and contains 0 and 1. These properties make Z an example
of a subring of C. A large class of subrings of C can be constructed as follows. For every
integer d € Z - {0}, the polynomial X²-d has exactly two roots in C (by the fundamental
theorem of algebra!): if we denote one of the roots by √d, then the other is -√d. Define
the subset Z[√d] of C by
Z[√d] = {z € C: z = a +b√d for some a, b = Z}.
The notation Z[√d] is read "Z adjoin √d." We use this terminology because (it turns out)
Z[√d is the smallest subring of C that contains Z and √d. By definition, the elements
of Z[√d] are the complex numbers having a particular form: a+b√d with a, b € Z. This
means that, to prove that a complex number is an element of Z[√d], we must show that the
number can be expressed in the indicated form. Keep this in mind when solving the first
problem below.
1. Let de Z-{0}.
(a) Using the definition of Z[√d], explain why Z is a subset of Z[√d].
(b) Prove that Z[√d] is closed under the addition operation on C.
(c) Prove that Z[√d] is closed under the multiplication operation on C.
(You will not actually need the assumption that d 0 in your arguments; but if d = 0,
then Z[√d] = Z, and we are most interested in the case where √d & Q, as this ensures that
Z[√d properly contains Z.)
2. Let d € Z - {0} and assume that √d & Q (we will eventually prove that this is true if
and only if d is not the square of an integer, i.e., if and only if √d & Z). In this case, the
elements of Z[√d are called quadratic integers.
(a) Prove that, if z € Z[√d], then there are unique integers a and b such that z = a+b√d.
(Hint: Such integers exist by the definition of Z[√d]. Assuming that there are integers
a₁, a2, b₁, and by such that z = a₁ + b₁ √d and z= a₂ + b₂√d, we need to show that
0₁ 0₂ and b₁ b₂. Ignoring the first equality, suppose for the sake of a contradiction
that b₁ b₂. Use the irrationality of √d to reach a contradiction. How does the
desired result follow?)
(b) We would like to define a function
=
: Z[√d] → Z[√d] by
a(z) = a - b√d,
where a, b € Z satisfy 2 = a +b√d. Why is there a potential issue with trying to
define a function using this formula? Explain how (a) resolves this issue, so that the
displayed formula does indeed give a well-defined function, which we call conjugation.
(Hint: compare with the first problem of Homework 1.)
(c) Prove that the function o defined in (b) is bijective.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 2 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
