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PreTeXt Sample Book Abstract Algebra (SAMPLE ONLY)

Thomas W. Judson, Isaac Newton (Editor)

Appendix B Hints and Answers to Selected Odd Exercises

2 The Integers
2.4 Exercises

2.4.1.

Answer.
The base case, S(1):[1(1+1)(2(1)+1)]/6=1=12 is true.
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Assume that S(k):12+22+⋯+k2=[k(k+1)(2k+1)]/6 is true. Then
12+22+⋯+k2+(k+1)2=[k(k+1)(2k+1)]/6+(k+1)2=[(k+1)((k+1)+1)(2(k+1)+1)]/6,
and so S(k+1) is true. Thus, S(n) is true for all positive integers n.
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2.4.3.

Answer.
The base case, S(4):4!=24>16=24 is true. Assume S(k):k!>2k is true. Then (k+1)!=k!(k+1)>2k⋅2=2k+1, so S(k+1) is true. Thus, S(n) is true for all positive integers n.
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2.4.11.

Hint.
The base case, S(0):(1+x)0−1=0≥0=0⋅x is true. Assume S(k):(1+x)k−1≥kx is true. Then
(1+x)k+1−1=(1+x)(1+x)k−1=(1+x)k+x(1+x)k−1≥kx+x(1+x)k≥kx+x=(k+1)x,
so S(k+1) is true. Therefore, S(n) is true for all positive integers n.
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2.4.19.

Hint.
Use the Fundamental Theorem of Arithmetic.
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2.4.23.

Hint.
Let S={s∈N:a∣s, b∣s}. Then S≠∅, since |ab|∈S. By the Principle of Well-Ordering, S contains a least element m. To show uniqueness, suppose that a∣n and b∣n for some n∈N. By the division algorithm, there exist unique integers q and r such that n=mq+r, where 0≤r<m. Since a and b divide both m, and n, it must be the case that a and b both divide r. Thus, r=0 by the minimality of m. Therefore, m∣n.
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2.4.27.

Hint.
Since gcd(a,b)=1, there exist integers r and s such that ar+bs=1. Thus, acr+bcs=c. Since a divides both bc and itself, a must divide c.
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2.4.29.

Hint.
Every prime must be of the form 2, 3, 6n+1, or 6n+5. Suppose there are only finitely many primes of the form 6k+5.
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3 Groups
3.5 Exercises

3.5.1.

Hint.
(a) 3+7Z={…,−4,3,10,…}; (c) 18+26Z; (e) 5+6Z.
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3.5.15.

Hint.
There is a nonabelian group containing six elements.
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3.5.17.

Hint.
The are five different groups of order 8.
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3.5.25.

Hint.
(aba−1)n=(aba−1)(aba−1)⋯(aba−1)=ab(aa−1)b(aa−1)b⋯b(aa−1)ba−1=abna−1.
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3.5.31.

Hint.
Since abab=(ab)2=e=a2b2=aabb, we know that ba=ab.
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3.5.35.

Hint.
H1={id}, H2={id,ρ1,ρ2}, H3={id,μ1}, H4={id,μ2}, H5={id,μ3}, S3.
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3.5.41.

Hint.
The identity of G is 1=1+02. Since (a+b2)(c+d2)=(ac+2bd)+(ad+bc)2, G is closed under multiplication. Finally, (a+b2)−1=a/(a2−2b2)−b2/(a2−2b2).
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3.5.49.

Hint.
Since a4b=ba, it must be the case that b=a6b=a2ba, and we can conclude that ab=a3ba=ba.
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5 Runestone Testing
5.8 True/False Exercises

5.8.1. True/False.

Hint.
Pn, the vector space of polynomials with degree at most n, has dimension n+1 by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter n?
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5.9 Multiple Choice Exercises

5.9.1. Multiple-Choice, Not Randomized, One Answer.

Hint 1.
What did you see last time you went driving?
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Hint 2.
Maybe go out for a drive?
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5.9.3. Multiple-Choice, Not Randomized, Multiple Answers.

Hint.
Do you know the acronym…ROY G BIV for the colors of a rainbow, and their order?
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5.9.5. Multiple-Choice, Randomized, One Answer but with Checkboxes.

Hint 1.
What did you see last time you went driving?
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Hint 2.
Maybe go out for a drive?
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5.9.7. 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.
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5.10 Parsons Exercises

5.10.1. Parsons Problem, Mathematical Proof.

Hint.
Dorothy will not be much help with this proof.
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5.12 Matching Exercises

5.12.3. Cardsort Problem, Linear Algebra.

Hint.
For openers, a basis for a subspace must be a subset of the subspace.
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5.13 Clickable Area Exercises

5.13.3. Clickable Areas, Text in a Table.

Hint.
Python boolean variables begin with capital latters.
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5.19 Fill-In Exercises

5.19.7. Fill-In, Dynamic Math with Formulas as Answers.

5.21 Exercises that are Timed

Timed Exercises

5.21.1. True/False.
Hint.
Pn, the vector space of polynomials with degree at most n, has dimension n+1 by Theorem 3.2.16. [Cross-reference is just a demo, content is not relevant.] What happens if we relax the defintion and remove the parameter n?
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5.28 Group Exercises

5.28.1. Multiple-Choice, Not Randomized, One Answer.

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