 # PreTeXt Sample Book: Abstract Algebra (SAMPLE ONLY)

## AppendixBHints and Answers to Selected Odd Exercises

View Source

### IBasics2The Integers2.4Exercises

#### 2.4.1.

The base case, $$S(1): [1(1 + 1)(2(1) + 1)]/6 = 1 = 1^2$$ is true.
Assume that $$S(k): 1^2 + 2^2 + \cdots + k^2 = [k(k + 1)(2k + 1)]/6$$ is true. Then
\begin{align*} 1^2 + 2^2 + \cdots + k^2 + (k + 1)^2 & = [k(k + 1)(2k + 1)]/6 + (k + 1)^2\\ & = [(k + 1)((k + 1) + 1)(2(k + 1) + 1)]/6\text{,} \end{align*}
and so $$S(k + 1)$$ is true. Thus, $$S(n)$$ is true for all positive integers $$n\text{.}$$

#### 2.4.3.

The base case, $$S(4): 4! = 24 \gt 16 =2^4$$ is true. Assume $$S(k): k! \gt 2^k$$ is true. Then $$(k + 1)! = k! (k + 1) \gt 2^k \cdot 2 = 2^{k + 1}\text{,}$$ so $$S(k + 1)$$ is true. Thus, $$S(n)$$ is true for all positive integers $$n\text{.}$$

#### 2.4.11.

Hint.
The base case, $$S(0): (1 + x)^0 - 1 = 0 \geq 0 = 0 \cdot x$$ is true. Assume $$S(k): (1 + x)^k -1 \geq kx$$ is true. Then
\begin{align*} (1 + x)^{k + 1} - 1 & = (1 + x)(1 + x)^k -1\\ & = (1 + x)^k + x(1 + x)^k - 1\\ & \geq kx + x(1 + x)^k\\ & \geq kx + x\\ & = (k + 1)x\text{,} \end{align*}
so $$S(k + 1)$$ is true. Therefore, $$S(n)$$ is true for all positive integers $$n\text{.}$$

#### 2.4.19.

Hint.
Use the Fundamental Theorem of Arithmetic.

#### 2.4.23.

Hint.
Let $$S = \{s \in {\mathbb N} : a \mid s\text{,}$$ $$b \mid s \}\text{.}$$ Then $$S \neq \emptyset\text{,}$$ since $$|ab| \in S\text{.}$$ By the Principle of Well-Ordering, $$S$$ contains a least element $$m\text{.}$$ To show uniqueness, suppose that $$a \mid n$$ and $$b \mid n$$ for some $$n \in {\mathbb N}\text{.}$$ By the division algorithm, there exist unique integers $$q$$ and $$r$$ such that $$n = mq + r\text{,}$$ where $$0 \leq r \lt m\text{.}$$ Since $$a$$ and $$b$$ divide both $$m\text{,}$$ and $$n\text{,}$$ it must be the case that $$a$$ and $$b$$ both divide $$r\text{.}$$ Thus, $$r = 0$$ by the minimality of $$m\text{.}$$ Therefore, $$m \mid n\text{.}$$

#### 2.4.27.

Hint.
Since $$\gcd(a,b) = 1\text{,}$$ there exist integers $$r$$ and $$s$$ such that $$ar + bs = 1\text{.}$$ Thus, $$acr + bcs = c\text{.}$$ Since $$a$$ divides both $$bc$$ and itself, $$a$$ must divide $$c\text{.}$$

#### 2.4.29.

Hint.
Every prime must be of the form 2, 3, $$6n + 1\text{,}$$ or $$6n + 5\text{.}$$ Suppose there are only finitely many primes of the form $$6k + 5\text{.}$$

### IIAlgebra1Groups1.5Exercises

#### 1.5.1.

Hint.
(a) $$3 + 7 \mathbb Z = \{ \ldots, -4, 3, 10, \ldots \}\text{;}$$ (c) $$18 + 26 \mathbb Z\text{;}$$ (e) $$5 + 6 \mathbb Z\text{.}$$

#### 1.5.15.

Hint.
There is a nonabelian group containing six elements.

#### 1.5.17.

Hint.
The are five different groups of order 8.

#### 1.5.25.

Hint.
\begin{align*} (aba^{-1})^n & = (aba^{-1})(aba^{-1}) \cdots (aba^{-1})\\ & = ab(aa^{-1})b(aa^{-1})b \cdots b(aa^{-1})ba^{-1}\\ & = ab^na^{-1}\text{.} \end{align*}

#### 1.5.31.

Hint.
Since $$abab = (ab)^2 = e = a^2 b^2 = aabb\text{,}$$ we know that $$ba = ab\text{.}$$

#### 1.5.35.

Hint.
$$H_1 = \{ id \}\text{,}$$ $$H_2 = \{ id, \rho_1, \rho_2 \}\text{,}$$ $$H_3 = \{ id, \mu_1 \}\text{,}$$ $$H_4 = \{ id, \mu_2 \}\text{,}$$ $$H_5 = \{ id, \mu_3 \}\text{,}$$ $$S_3\text{.}$$

#### 1.5.41.

Hint.
The identity of $$G$$ is $$1 = 1 + 0 \sqrt{2}\text{.}$$ Since $$(a + b \sqrt{2}\, )(c + d \sqrt{2}\, ) = (ac + 2bd) + (ad + bc)\sqrt{2}\text{,}$$ $$G$$ is closed under multiplication. Finally, $$(a + b \sqrt{2}\, )^{-1} = a/(a^2 - 2b^2) - b\sqrt{2}/(a^2 - 2 b^2)\text{.}$$

#### 1.5.49.

Hint.
Since $$a^4b = ba\text{,}$$ it must be the case that $$b = a^6 b = a^2 b a\text{,}$$ and we can conclude that $$ab = a^3 b a = ba\text{.}$$

#### 1.5.55.

$$1$$

#### 1.5.57.

$$n$$

#### 1.5.59.

##### 1.5.59.a
$$2$$
##### 1.5.59.b
###### 1.5.59.b.i
$$6$$
###### 1.5.59.b.ii
$$10$$

### 3Runestone Testing3.7True/False Exercises

#### 3.7.1.True/False.

Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$

### 3.8Multiple Choice Exercises

#### 3.8.1.Multiple-Choice, Not Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?

Hint 1.
What did you see last time you went driving?
Hint 2.
Maybe go out for a drive?

#### 3.8.5.Mathematical Multiple-Choice, Not Randomized, Multiple Answers.

Hint.
You can take a derivative on any one of the choices to see if it is correct or not, rather than using techniques of integration to find a single correct answer.

### 3.9Parsons Exercises

#### 3.9.1.Parsons Problem, Mathematical Proof.

Hint.
Dorothy will not be much help with this proof.

### 3.11Matching Exercises

#### 3.11.3.Matching Problem, Linear Algebra.

Hint.
For openers, a basis for a subspace must be a subset of the subspace.

### 3.12Clickable Area Exercises

#### 3.12.3.Clickable Areas, Text in a Table.

Hint.
Python boolean variables begin with capital latters.

### 3.16Hodgepodge>

#### 3.16.1.With Tasks in an Exercises Division.

##### 3.16.1.aTrue/False.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$

### 3.17Exercises that are Timed

#### Timed Exercises

##### 3.17.1.True/False.
Hint.
$$P_n\text{,}$$ the vector space of polynomials with degree at most $$n\text{,}$$ has dimension $$n+1$$ by Theorem 1.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter $$n\text{?}$$