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### Homework 3

- Section 5.5
- Homework: 1, 2, 6, 7, 8, 9, 11
- Hand In: 6, 7 (please refrain from using online resources)

- Section 7.1
- Homework: 5, 7, 9, 13, 23, 25 and the following problems:
- 31. Show that the set \(Q\) of all matrices of the form \[\left[ \begin{array}{cc}a+bi&c+di\\ -c+di&a-bi \end{array}\right],\] where \(i^2=-1\) and \(a,b,c,d \in \mathbb{R}\), is a ring with unity under the usual operations of matrix addition and multiplication (in fact, a subring of \(M_2(\mathbb{C})\)). Give an example to show that \(Q\) is not commutative. Then for \(A \in Q\) compute \(A\bar{A}\) and \(\bar{A}A\), where \[\overline{\left[ \begin{array}{cc}a+bi&c+di\\ -c+di&a-bi \end{array}\right]} = \left[ \begin{array}{cc}a-bi&-c-di\\ c-di&a+bi \end{array}\right].\] Hence determine which elements of \(Q\) are invertible and determine a formula for the inverse of each such element. (Adapted from: Charles C. Pinter, "A Book of Abstract Algebra", Dover, Mineola, NY (2010))
- 32. An element \(a\) of a ring is
*unipotent*if \(1-a\) is nilpotent (see Exercise 13 of Section 7.1 for the definition of nilpotent element of a ring). Let \(R\) be a commutative ring and suppose \(a, b \in R\) are both unipotent. Show that \(ab\) is unipotent. Show also that every unipotent element of a ring is invertible. (Source: Charles C. Pinter, "A Book of Abstract Algebra", Dover, Mineola, NY (2010))

- Hand In: 31, 32

- Homework: 5, 7, 9, 13, 23, 25 and the following problems:
- Section 7.3
- Homework: 1, 8, 12, 22, 24, 29
- Hand In: 22, 29 (please refrain from using online resources)

- Section 7.4
- Homework: 7, 13, 16, 30, 31, 32
- Hand In: 30, 32 (please refrain from using online resources)

- Section 7.6
- Homework: 1, 2, 5 and the following problem:
- 12. Let \(R = \mathbb{Z}, I_1 = 6\mathbb{Z}\) and \(I_2 = 4\mathbb{Z}\). Show that the map \(\psi\) of Theorem 10.1 (class notes) is not surjective. (Source: Thomas W. Hungerford, "Algebra", Springer, New York, NY (1980, revised eighth printing 1996))

- Hand In: 12

- Homework: 1, 2, 5 and the following problem:
- Section 9.1
- Homework: 4, 9, 11, 12 and the following problems:
- 19. (Evaluation or substitution homomorphism) Suppose \(R\) and \(S\) are commutative rings and \(s_1,\ldots, s_n \in S\). Let \(\varphi : R \to S\) be a ring homomorphism. Show that the map \(\psi : R[x_1, \ldots, x_n] \to S\) given by \(\psi(f(x_1,\ldots, x_n)) = \varphi(f(s_1,\ldots, s_n))\) is a ring homomorphism. (Adapted from: Thomas W. Hungerford, "Algebra", Springer, New York, NY (1980, revised eighth printing 1996))
- 20. Let \(R\) be the ring \(M_2(\mathbb{Z})\) of \(2\times 2\) matrices with integer entries. Show that there exist \(C, A \in R\) such that \((C-A)(C+A) \neq C^2 - A^2\). Hence show that the map \(\psi\) of Problem 19 need not be a ring homomorphism if one of the rings involved is not commutative (consider the polynomial \(x^2 - A^2 \in M_2(\mathbb{Z})[x])\). (Adapted from: Thomas W. Hungerford, "Algebra", Springer, New York, NY (1980, revised eighth printing 1996))

- Hand In: 19, 20

- Homework: 4, 9, 11, 12 and the following problems:
- Section 8.2
- Homework: 3, 4, 5, 7
- Hand In: none

- Section 8.3
- Homework: 1, 5, 8 and the following problem:
- 12. Let \(R\) be the subring \(\{a+b\sqrt{10}\ |\ a,b \in \mathbb{Z}\}\) of the field of real numbers. Show the following:
- The map \(N\ :\ R \to \mathbb{Z}\) given by \(a+b\sqrt{10} \mapsto (a+b\sqrt{10})(a-b\sqrt{10}) = a^2 - 10b^2\) is such that \(N(uv) = N(u)N(v)\) for all \(u, v \in R\) and \(N(u) = 0\) if and only if \(u=0\).
- \(u\) is a unit in \(R\) if and only if \(N(u)=\pm 1\).
- \(2, 3, 4+\sqrt{10}\) and \(4-\sqrt{10}\) are irreducible elements of \(R\).
- \(2, 3, 4+\sqrt{10}\) and \(4-\sqrt{10}\) are not prime elements of \(R\).

- 12. Let \(R\) be the subring \(\{a+b\sqrt{10}\ |\ a,b \in \mathbb{Z}\}\) of the field of real numbers. Show the following:
- Hand In: 12

- Homework: 1, 5, 8 and the following problem: