## Math 5110 Graduate Algebra I

### Homework 3

Due Wednesday October 24, in class (individually)
• 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
• 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
• 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
• 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:
1. 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 $$u=0$$.
2. $$u$$ is a unit in $$R$$ if and only if $$N(u)=\pm 1$$.
3. $$2, 3, 4+\sqrt{10}$$ and $$4-\sqrt{10}$$ are irreducible elements of $$R$$.
4. $$2, 3, 4+\sqrt{10}$$ and $$4-\sqrt{10}$$ are not prime elements of $$R$$.
(Source: Thomas W. Hungerford, "Algebra", Springer, New York, NY (1980, revised eighth printing 1996))
• Hand In: 12