Problem 1 The subset Z of C is closed under the addition and multiplication operations on C; it isalso closed under subtraction, and contains 0 and 1. These properties make Z an exampleof a subring of C. A large class of subrings of C can be constructed as follows. For everyinteger d € Z - {0}, the polynomial X²-d has exactly two roots in C (by the fundamentaltheorem of algebra!): if we denote one of the roots by √d, then the other is -√d. Definethe subset Z[√d] of C byZ[√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 elementsof Z[√d] are the complex numbers having a particular form: a+b√d with a, b € Z. Thismeans that, to prove that a complex number is an element of Z[√d], we must show that thenumber can be expressed in the indicated form. Keep this in mind when solving the firstproblem 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 thatZ[√d properly contains Z.)2. Let d € Z - {0} and assume that √d & Q (we will eventually prove that this is true ifand only if d is not the square of an integer, i.e., if and only if √d & Z). In this case, theelements 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 integersa₁, a2, b₁, and by such that z = a₁ + b₁ √d and z= a₂ + b₂√d, we need to show that0₁ 0₂ and b₁ b₂. Ignoring the first equality, suppose for the sake of a contradictionthat b₁ b₂. Use the irrationality of √d to reach a contradiction. How does thedesired result follow?)(b) We would like to define a function=: Z[√d] → Z[√d] bya(z) = a - b√d,where a, b € Z satisfy 2 = a +b√d. Why is there a potential issue with trying todefine a function using this formula? Explain how (a) resolves this issue, so that thedisplayed 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.

Answered: Image /qna-images/answer/00481e6c-fc35-43e5-a7f1-55c2f4f9a880.jpg

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.