Problem 3 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 \( d \in \mathbb{Z} - \{0\} \), the polynomial \( X^2 - d \) has exactly two roots in ℂ (by the fundamental theorem of algebra): if we denote one of the roots by \( \sqrt{d} \), then the other is \( -\sqrt{d} \). Define the subset \( \mathbb{Z}[\sqrt{d}] \) of ℂ by\[ \mathbb{Z}[\sqrt{d}] = \{z \in \mathbb{C} : z = a + b\sqrt{d} \text{ for some } a, b \in \mathbb{Z}\}. \]The notation \( \mathbb{Z}[\sqrt{d}] \) is read “ℤ adjoin \( \sqrt{d} \).” We use this terminology because (it turns out) \( \mathbb{Z}[\sqrt{d}] \) is the smallest subring of ℂ that contains ℤ and \( \sqrt{d} \). By definition, the elements of \( \mathbb{Z}[\sqrt{d}] \) are the complex numbers having a particular form: \( a + b\sqrt{d} \) with \( a, b \in \mathbb{Z} \). This means that, to prove that a complex number is an element of \( \mathbb{Z}[\sqrt{d}] \), 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 \( d \in \mathbb{Z} - \{0\}.** (a) Using the definition of \( \mathbb{Z}[\sqrt{d}] \), explain why ℤ is a subset of \( \mathbb{Z}[\sqrt{d}] \). (b) Prove that \( \mathbb{Z}[\sqrt{d}] \) is closed under the addition operation on ℂ. (c) ### Problem Statement:3. For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample.(a) If \(d \in \mathbb{Z} - \{0\}\) and \(a_1, a_2, b_1, b_2 \in \mathbb{Z}\), then \(a_1 + b_1 \sqrt{d} = a_2 + b_2 \sqrt{d}\) if and only if \(a_1 = a_2\) and \(b_1 = b_2\).(b) If \( R \) is a ring and \( x, y \in R^x \), then \( xy \in R^x \).^2

Answered: Image /qna-images/answer/3c4f0a92-f18d-4359-bc7a-060ba14332d5.jpg

3. For each of the following statements, indicate whether the statement is true or false and justify your answer with a proof or a counterexample. (a) If d = Z-{0} and a₁, a2, b₁,b₂ € Z, then a₁ +b₁√d=a₂+ b₂√d if and only if a₁ = 02 and b₁ b₂. (b) If R is a ring and r, y € R*, then xy € RX.²