# Herstein Homework Solutions

## Spring 2016

Lecture: Tuesday, Thursday 1:30-2:45, in Keller 402
Instructor:Ralph Freese
Office: 305 Keller
email:
Office hours: T-Th 2:45-3:10 and by appointment (or just come to my office
and see if I'm in)

• Project Paper and Presentation. Besides the web and wikipedia in particular, a good source is Keith Conway's page.

• Sample paper on cyclotomic polynomials:

• Finite Fields: This project should involve two or maybe three people working together. The first part should cover the basic properties (the book covers these pretty well). The second should talk about actually constructing finite fields. Here are some slides that have a good explanation: Finite Fields.

Conway Polynomials should also be discussed. The wikipedia page is pretty good. Also Frank Luebeck page does a nice job and actually gives many of them.

• Topics in Ring Theory: For this project you could choose something from ring theory, possibly something from Chapter 10. Or you could cover Gaussian integers and Eisenstein integers. Gaussian integers is just the ring $$\mathbb Z[i] = \{a + bi : a, b \in \mathbb Z\}$$. While Eisenstein integers are the ring $$E = \{a + b\omega : a, b \in \mathbb Z\}$$, where $$\omega = (-1 + \sqrt 3i)/2$$. Both have a norm which helps in understanding the arithmetic of the rings. For Gauassian integers it is ($$N(a+ib) = a^2 + b^2$$ which is positive-definite. But the norm for Eisenstein is not positive-definite, which makes things interesting. Here is a sample paper with questions you can use to get started.

• Straight-edge and compass constructions: Basically a length $$a$$ can be constucted (from a unit length) iff $$[\mathbb Q(a):\mathbb Q] = 2^k$$ for some $$k$$. You should outline the proof of this and use to show that doubling a cube, trisecting an angle, and squaring the circle are all impossible. You should also look into which regular polygons can be constructed. At least show a pentagon can be constructed and a septagon cannot. The key is to find the minimum polynomial of $$\zeta_n + \zeta_n^{-1}$$, where $$\zeta_n = e^{2\pi i/n}$$ defined in the cyctomic paper above.

• Group Action and Sylow's Theorem:Group Actions.

• Book: T. Hungerford, Abstract Algebra: An Introduction, third edition.

• Other Good Books:
• Herstein, Topics in Algebra.

• $${\rm \TeX}$$ (and $${\rm \LaTeX}$$): is a software system for typing mathematics. You will be required to use this for the homework.
• Homework: Download the first two files and change the file to have your name. In the file change "Billy Bob" to your name. The starred problems are to be written up very carefully, using complete sentences. (Afterall this is a writing intensive course.) Future assignments will be on the Homework page.

The first two books are considered easier'' books, and although they also have a somewhat different approach, they have most of the topics we will cover and may be of good help if you have difficulty reading Artin's book.

The last one is a standard'' text for a first course in abstract algebra, but might have a higher level of difficulty. (It's been used for the honors section of this course.) Nevertheless, it is a very good reference.

### Legal Issues

#### Conduct

All students should be familiar and maintain their Academic Integrity'': from Hilltopics 2006/2007, pg.40:

The responsibility for learning is an individual matter. Study, preparation and presentation should involve at all times the student's own work, unless it has been clearly specified that work is to be a team effort. Academic honesty requires that all work presented be the student's own work, not only on tests, but in themes, papers, homework, and class presentation. There is a clear distinction between learning new ideas and presenting them as facts or as answers, and presenting them as one's own ideas. It is part of the learning process to incorporate the thoughts or ideas of others into one's own mind and presentations with the purpose of learning and enlarging on personal boundaries of knowledge.

You should also be familiar with the Classroom Behavior Expectations''.

#### Disabilities

Students with disabilities that need special accommodations should contact the Office of Disability Services'' and bring me the appropriate letter/forms.

#### Sexual Harassment and Discrimination

For Sexual Harassment and Discrimination information, please visit the Office of Equity and Diversity''.

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### Important dates:

Wednesday, August 23 - Classes begin.

Friday, September 01 - Last day to add, change grade options, or drop a full semester course without a "W".

Monday, September 04 - Labor Day Holiday. (No class.)

Friday, September 29 - Midterm I.

Tuesday, October 03 - Last day to drop a full term course with a "W".

Thursday-Friday, October 12-13 - Fall Break. (No class.)

Monday, November 06 - Midterm II.

Tuesday, November 14 - Last day to drop with a WP/WF.

Thursday-Friday, November 23-24 - Thanksgiving. (No class.)

Tuesday, December 05 - Last Class Day.

Friday, December 08 - Final.

More dates.

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### Solutions to Selected HW Problems

Please read: I will try to post here a few solutions. The new solutions will be added to this same file. They might come with no explanation, just the answer''. If yours do not match mine, you can try to figure out again. (Also, read the disclaimer below!) You can come to office hours if you want explanations for the answers. Be careful that just because our answers'' were the same, it doesn't mean that you solved the problem correctly (it might have been a fortunate'' coincidence), and in the exams what matters is the solution itself. I will do my best to post somewhat detailed solutions, though.

Disclaimer: I will have to put these solutions together rather quickly, so they are subject to typos and conceptual mistakes. (I expect you to be a lot more careful when doing your HW than I when preparing these.) You can contact me if you think that there is something wrong and I will fix the file if you are correct.

Solutions to Selected HW Problems (Click on Refresh'' or Reload'' if you don't see the changes!)

CHANGE LOG:

• (12/01 -- 5:40pm) Added solutions of the extra credit problems from HW14 and for a few problems from HW15.
UPDATE: Since 10.3 is not in the exam, this update was the final one!
• (11/29 -- 3:30pm) Added solutions of HW14.
• (11/23 -- 10:05am) Added solutions of HW13.
• (11/17 -- 11:40am) Added solutions of HW12 (and fixed some typos in the previous HWs).
• (11/09 -- 1:45pm) Added solutions of HW11.
• (11/02 -- 11:20am) Added solutions of HW10.
• (10/27 -- 4:45pm) Added solutions of HW9.
• (10/18 -- 11:30am) Added solutions of HW8.
• (10/16 -- 11:45am) Added solutions of HW7.
• (10/06 -- 11:30am) Added solutions of HW6.
• (09/27 -- 2:30pm) Added solutions of HW5.
• (09/22 -- 9:30am) Added solutions of HW4.
• (09/14 -- 7:00pm) Added solutions of HW3.
• (09/07 -- 4:35pm) Posted solutions of HW1 and HW2.

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### Homework

HW1 - Due on Monday 08/28:

Review matrices and vector spaces (from Math 251/257).

Section 1.1 (pg. 31): 1, 2, 4, 11.

Section 1.2: 2, 12, 14.

Section 1.3: 1(b), (c), (d), 2 (Hint: row operations).

HW2 - Due on Wednesday 09/06:

Section 1.4: 1, 4.

Section 2.1 (pg. 69): 1(b), 3, 4, 7, 11. (Note: I did not do in class anything very similar to number 4, but just remember that every element of a group has an inverse. It is almost like you can divide'' both sides by something, but in fact you multiply both sides by an inverse to get rid of a term. But be careful: the group might not be Abelian!

HW3 - POSTPONED!!! (See Important Notes above.) Due on Wednesday 09/13:

Section 2.2: 2, 3, 4, 11, 13, 16(a), 17. Note: I excluded number 10 (from my previous list here)!

HW4 - POSTPONED!!! (See Important Notes above.) Due on Wednesday 09/20:

Section 2.3: 1, 4, 5, 9, 12.

Note: For 12 you need to know what an automorphism is: an automorphism is just an isomorphism from a group G onto itself.

HW5 - Due on Wednesday 09/27:

Section 2.4: 4, 8(b), 9, 13(a), 14, 17, 22.

HW6 - Due on Wednesday 10/04:

Section 2.5: 2, 5, 6. (Maybe you should also review a little about equivalence relations from Math 300.)

HW7 - Due on Wednesday 10/11:

Section 2.6: 1, 3, 5, 10(a). (Hint for 10(a): use Proposition 6.18)

Section 2.7: 1, 3.

HW8 - Due on Wednesday 10/18:

Section 2.8: 2, 3, 4(a), 4(b), 5, 6.

HW9 - Due on Wednesday 10/25:

Section 2.9: 1, 2, 5.

Section 2.10: 1(a), (b), (d), 3, 9, 10. (The last one was supposed to be 11.)

HW10 - Due on Wednesday 11/01:

Section 5.5 (pg. 192): 1(b), (c), 3, 8 (hint for 8(a)), 11(a).

Section 5.6: 1.

You can also try the Extra Credit Problem, but it is due in class Monday (10/30), NOT Wednesday!!!You are not allowed to discuss this problem with anyone!

HW11 - Due on Wednesday 11/08:

Section 5.6: 2, 6, 7.

Section 5.7: 6. (Hints: For (a) use problem 5: you can assume its statement is true without proving it. For (b), don't make it complicated. Look at some of our most used'' groups.)

HW12 - Due on Wednesday 11/15:

Section 6.1 (pg. 229): 1, 2, 4, 6, 10(a), (e), 12.

Section 6.3: 4, 5, 7, 13.

HW13 - Due on Wednesday 11/22:

Section 6.4: 1, 2, 3, 13, 15.

HW14 - Due on Wednesday 11/29:

Section 5.3 (pg. 189): 1, 2, 4. (These are about the dihedral group. Note that the text uses D_n for what we use D_2n.) Hint for 4(c): The answer is yes and to prove it you just need to show that the product group has two generators that satisfy the same properties as the two generators of the dihedral group.

Section 6.7 (pg. 233): 1.

Section 6.8: 1 (to show this, you basically need to show that you can get a and b from the two elements given in the problem), 6 (there is a typo in the statement: it should read N and G/N cyclic'' instead of G and G/N cyclic''), 16.

Extra Credit: Prove that if a group has order 24, then it is not simple. Alternatively, prove the same for order 36. (You can do either one.) These might be difficult... (Or, maybe I just did not see the easy way...) You are not allowed to discuss this problem with anyone! You should not look for the solved problem in other references either!

HW15 - not to turn in (just to study for the final):

Section 10.1 (pg. 379): 6 (you can replace the complex numbers with real numbers in this problem), 8(a), (c), 9(a), (c), 11(a), (b), 13.

Section 10.3:NOT IN THE EXAM! (Do it only if you want to have some fun...) 2, 4, 8(a), 9, 15, 17, 19, 20, 22, 29, 30.

PLEASE, HIT REFRESH'' (OR RELOAD'') IN YOUR BROWSER WHEN VISITING THIS PAGE!!!!!!! I usually get messages asking for the update in the HW when it has already been updated. Since I change this page often, some times the browser don't see the changes. But, if you hit refresh and there is still problems missing, feel free to write me.

If it is already Friday afternoon and there still is a More to come'' after the HW assignment due on the coming Monday, write me an e-mail at lfinotti@utk.edu, and I'll update it and let you know.

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