# Abstract Algebra Homework Solutions Birkhoff

This course is an introduction to groups and rings, which are foundational concepts in modern mathematics. Along the way, we will also deepen our understanding of linear algebra and the role of structures on vector spaces.

Prerequisites: I will assume you are all familiar with real vector spaces, linear transformations, and matrices. Basic familiarity with mathematical proof is necessary; I recommend that you have taken at least one proof-based class before.

**Announcements**

Tue, Sep 12. Office hours and sections are being finalized. Follow the link for the schedule.

Tue, Sep 12. See announcement about new room location: Northwest building Room B101. This applies for tomorrow (Wed, Sep 13).

Mon, Sep 11. See announcement about clarifications on optional problems for HW 2.

Sat, Sep 9. Please see announcement about how to submit homeworks.

Friday Sept 1, 10 PM: There was ambiguity in problem 3(b); there, "id_Z" means the identity function from Z to itself. It does NOT mean "identity element of Z." The updated pdf file for the homework reflects this in blue font.

Friday Sept 1, 10 PM: The notes from the past rendition of my Math 122 class are here.

Friday Sept 1, 10 PM: Please be patient as your CAs and I figure out the best way to split up the grading. It seems easiest to break up the assignments problem by problem, so each CA can grade a different problem without crossing wires. It's okay if your uploaded PDFs/images contain superfluous problems (e.g., if what you upload to "Problem Two" also includes your solutions to Problems One or Three, for instance).

Your first two homework assignments are posted: One assignment is a survey, the other is a problem set. Both are due by Wed, Sep 6.

Finally, feel free to check out Beyond this course , where I have small links to topics/history/ideas that this course has influenced, or has been influenced by.

**Lectures, Notes, and Resources**

**You can ask in-class questions, or questions in general, here.**

Here is a running list of past and upcoming lecture topics (in case you want to read ahead in some of the suggested sources in the syllabus). Chances are I will not cover every topic I want to cover in the next lecture, so topics will start bleeding from one lecture to the next. (Hence only a seven-day forecast at most.)

- Wed, Aug 30. First Class. Groups as symmetries. Definitions groups, homomorphisms and isomorphisms. Examples: Z, R, R-0, GLn(R), Sn. Notes by Nat. See Artin Chapter 2.
- Fri, Sep 1. Basic facts about groups. Group tables. Z/nZ, Abelian-ness, mattress group. [Not covered; will cover later: Basic facts about homomorphisms, subgroups, subgroups of Z. Cyclic groups. Cyclic notation.] See Artin Chapter 2. Notes by Nat.Addendum from Hiro.
- Wed, Sep 6. Group actions. Orbits. More examples as needed; perhaps counting formula. (Toward Orbit-stabilizer.) See Artin 6.7. Notes by Nat.
- Fri, Sep 8. Hiro is out of town; guest lecture. Equivalence relations, equivalence classes. Challenge problems. Here is the hand-out. See Artin 2.7.
**How was the exercise session? Fill out this survey!** - Mon, Sep 11. Lagrange's Theorem. Normal subgroups and quotient groups. Artin 2.8 and 2.12 and 6.7. Nat's Notes. [Not covered: All topics listed.]
- Wed, Sep 13. Normal subgroups and quotient groups. Artin 1.5. Notes by Nat. [Topics not covered: Symmetric group. Cyclic notation. Cyclic groups. p-groups.]
- Fri, Sep 15. Quotient groups. Kernels, images, first isomorphism theorem, universal property of quotient groups. [Topics not covered: Conjugation action. Class equation. Artin 7.2.] Notes by Nat.
- Mon, Sep 18. First isomorphism theorem, universal property of quotient groups. Cyclic groups. Subgroup generated by a single element, order of elements. Classification fo cyclic groups. [Not covered: Centers. Commutator subgroup. Abelianizations.] (Artin 7.10, 2.12, 2.4) Notes by Nat.
- Wed, Sep 20. Dihedral group. Generators and relations. (Artin 7.2, exercises for 7.10) [Topics not covered: S_n, Centers, commutator subgroups, abelianizations. Quaternions, dihedral groups. Conjugation action. Class equation.] Notes by Nat.
- Fri, Sep 22. S_n and A_n. Sign. Subgroups and diagrams of subgroups. Conjugation in S_n. (Artin 7.5, 1.5, 2.2)
- Mon, Sep 25. Conjugation in S_n. Cayley's Theorem. Abelianizations, commutator subgroups. [Topics not covered: Short exact sequences. Classifying finite groups. Simple groups.] (Artin 2.4, 6.1, exercises for 6.8)
- Wed, Sep 27. More on quotients; what it means to generate; forget cosets. [Not covered: A_n. Index. Automorphisms of groups. Conjugation action. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups.] (Special Homework Assignment: 20 Questions for Groups.)
- Fri, Sep 29. Conjugation. Automorphisms of groups. Class equation. Orbit-stabilizer theorem, counting. Centers of p-groups. Cauchy's Theorem, index (i.e., applications of counting).[Artin 6.1]
- Mon, Oct 2. Midterm One given out. Example Week begins: Fundamental groups.
- Wed, Oct 4. Example week continues: Elliptic curves.
- Fri, Oct 6. Example week continues: Subgroup of O(3), or symmetries of regular polyhedra in R^3.

- Wed, Oct 11. Midterm due. Deadline for Classroom-to-table. Subgroups of SO(3).
- Fri, Oct 13. Equivalence relations. Introduction to rings.
- Mon, Oct 16. Fields, units. Z/nZ. More rings. Endomorphisms of abelian groups as rings. [not covered: Zero divisors.] (Artin Chapter 11)
- Wed, Oct 18. Ring homomorphisms. Ideals. Quotient rings. [not covered: Zero divisors. Fields.] (Artin Chapter 11)
- Fri, Oct 20. Z/nZ, Z/pZ, integral domains, fields, examples. Modules. Linear algebra over arbitrary fields. (Artin Chapters 11 and 14)
- Mon, Oct 23. Free modules. Submodules. Ideals are submodules. Maps of modules. Kernel, Cokernel, Image, Matrices. Determinants. Generation of modules. Generation of ideals.
- Wed, Oct 25. Adjugate matrices; invertible linear transformations; Cayley-Hamilton Theorem. GL_n(F)
- Fri, Oct 27. Matrices are the same thing as endomorphisms of R^{\oplus n}. [Not covered: Polynomials with coefficients in fields. Integers. PIDs. Irreducibility. Euclidean algorithm for polynomials. Quotients of polynomial rings.]
- Mon, Oct 30. Matrices with unit determinants are invertible. [Not covered: Classification of finitely generated modules over PIDs. Classification of abelian groups.]
- Wed, Nov 1. Finitely generated modules over vector spaces are free. [Not covered: Back to groups. p-Sylow subgroups. Sylow theorems. Applications.]
- Fri, Nov 3. Cayley-Hamilton Theorem. [Not covered: Short exact sequences. Semidirect products.] Some notes about, including a proof of, the Cayley-Hamilton Theorem.
- Mon, Nov 6. (If we're not behind: Review session.)
- Wed, Nov 8. (If we're not behind: Review session.)
- Fri, Nov 10. In-class midterm.
- Mon, Nov 13. Sylow theorems.
- Wed, Nov 15. Applications of Sylow theorems. Cauchy's theorem.
- Fri, Nov 17. Semi-direct products.
- Mon, Nov 20. Statement of classification of finitely generated modules over PIDs.
- Mon, Nov 27. Application of classification of finitely generated modules over PIDs. Polynomials. Jordan Normal Form.
- Wed, Nov 29. Final Exam distributed. Lower central series. Why the classification of finite simple groups?
- Fri, Dec 1. Last Day of classes.

- Thu, Dec 14. Final Exam due.

- Sometime in the future: What can we do with integers? Rings, subrings of C, modules. (Artin Chapter 11)
- Direct products and semidirect products. Short exact sequences.
- Classification of finite groups. Simple groups. Jordan-Holder decomposition.

There are also live-TeXed notes taken by Daniel.

### Math 250A, Groups, Rings, and Fields

### Fall 2012

### MWF 1-2 PM, 85 Evans

Textbook Grading Homework Assignments Course Schedule Notes Back to Main Page

#### About the course:

Professor: Elena FuchsOffice: 851 Evans

Email: efuchs at math dot berkeley dot edu

Office hours: Mondays 2-4 and Wednesdays 2-3:30 or by appointment.

**ANNOUNCEMENT:**Our final will take place on Wednesday, Dec. 12, 7-10PM in 101 LSA (Life Sciences Addition). You are allowed to bring 8 pages of handwritten notes to the exam to use during the test.

**Please bring a blue book to record your solutions.**We will be doing some review (and some category theory) during RRR week during regular class time in 85 Evans. We will have usual office hours both during RRR week and during the week of the final. Here are some problems which might help you prepare for the final. Some more practice problems can be found here. It is of course also a good idea to go over the homework assignments and midterm in preparation for the exam. Good luck!

This course is meant to equip the student with a thorough background in Algebra. This includes, very roughly, various standard important theorems on groups, rings, modules, and fields.

The assumption is that students taking this course have a solid background in undergraduate algebra -- the official prerequisites for this course are Math 113 and 114. Another assumption is that students will read about the basic notions in the text, and digest these notions, as well as those covered in class, through weekly homework assignments. There is a lot of material to cover, and so class time will be devoted only to main theorems and applications thereof: for example, definitions which you should either have seen in undergraduate algebra or could easily grasp by reading the book will not be covered during class time. On the other hand, we will cover some material in class that is either not emphasized in the book or deferred to an exercise.

To make it easier for you to prepare for the lectures, I will post in advance a weekly list of concepts we will cover in class, as well as notions that I will assume you know throughout each lecture on this website.

**About the text:**

The textbook for this class is *Algebra* by S. Lang, from which we will cover much of chapters I-VI. It is the standard book to use for a first year graduate algebra course, perhaps because it contains all of the topics one might cover in such a course and much more. It will serve you not only as a textbook for this course but also as an algebra reference throughout your mathematical endeavors.

Many students find that this book is much too dense. If you feel this way, I recommend that, rather than reading from a different book completely, you supplement your reading from Lang with additional reading from your undergraduate algebra book. Some other books I could recommend are

*Algebra*by Artin*Algebra*by Birkhoff and MacLane*Abstract Algebra*by Dummit and Foote- J.S. Milne's notes on group theory and Galois theory on this website

Keep in mind that Lang's book is written with a fairly mature mathematical audience in mind. The proofs often require the reader to fill in the details, and the examples sometimes assume material which you are by no means required to know (although these examples are usually very interesting). It is not expected that you will understand all of these examples.

**Grading:**

Grades will be based on weekly homework assignments (50%), one take home midterm (15%), and one in class final (35%). Most homeworks (unless otherwise noted) will be due on Fridays in class or under my office door by 2PM. Collaboration on the homework is encouraged, but please do cite any sources which helped you to complete the assignment. ** No late homeworks will be accepted!** The midterm will be handed out in class on Friday, October 12th, and due in class on Friday, October 19th. You may not collaborate with anyone on the midterm: any academic dishonesty on the exam will result in a score of 0 for the exam. Our final is scheduled for December 12, 7-10PM.

**Supplementary notes:**

**Course schedule:**

- 8/24: Isomorphism theorems, exact sequences, composition series. This week's suggestions on what to know before class are here.
- 8/27: Composititon series and Jordan-Holder theorem.
- 8/29: Solvable groups.
- 8/31: Solvable groups continued, beginning group actions.
- 9/5: Group actions and automorphism groups. This week's suggestions on what to know before class are here.
- 9/7: Semidirect products and applications.
- 9/10: Sylow Theorems.
- 9/12: Sylow Theorems and applications.
- 9/14: Beginning rings: ideals and Chinese Remainder Theorem. This week's suggestions on what to know before class are here.
- 9/17: Chinese Remainder Theorem, applications thereof, beginning proof that every PID is a UFD.
- 9/19: Continuing proof that every PID is a UFD.
- 9/21: If D is a UFD then D[x] is a UFD, beginning Hilbert's Basis theorem.
- 9/24: Finishing Hilbert's Basis theorem, beginning Mason-Stothers Theorem and ABC conjecture (see IV.7).
- 9/26: Finishing Mason-Stothers Theorem, beginning fields. This week's suggestions on what to know before class are here.
- 9/28: Transcendence degree continued.
- 10/1: Composite fields.
- 10/3: Splitting fields and algebraic closure.
- 10/5: Proof of existence of algebraic closure, starting separable extensions.
- 10/8: Separable extensions, finite fields.
- 10/10: Finishing off finite fields, beginning Galois theory. This week's suggestions on what to know before class are here.
- 10/12: L-valued characters, Galois extensions.
- 10/15: Galois groups and equivalent definitions of Galois extensions.
- 10/17: Fundamental theorem of Galois theory.
- 10/19: Finishing proof of fundamental theorem of Galois theory and applications.
- 10/22: Applications of Galois correspondence. This week's suggestions on what to know before class are here.
- 10/24: Fundamental theorem of algebra, starting cyclic extensions with solvability of polynomials in mind.
- 10/26: Radical extensions and Galois groups of composite extensions.
- 10/29: Galois' criterion for solvability of polynomials.
- 10/31: Solving cubics and quartics.
- 11/2: Solving quartics continued, some impossible and possible geometric constructions.
- 11/5: Galois groups of quartics, constructing n-gons. You might want to read the notes on geometric constructions posted above.
- 11/7: Finishing off geometric constructions, beginning modules. This week's suggestions on what to know before class are here.
- 11/9: Submodules, quotient modules, direct sums of modules, torsion modules.
- 11/14: Noetherian modules, cyclic modules.
- 11/16: Beginning discussion of finitely generated torsion modules over a PID.
- 11/19: Structure theorem for finitely generated torsion modules over a PID.
- 11/21: Free modules and structure theorem for finitely generated modules over a PID.
- 11/26: Proof of structure theorem for finitely generated modules over a PID, Jordan Canonical Form.
- 11/28: Projective modules vs. free modules.
- 11/30: Some category theory, abelian categories.

**Homework assignments:**

Some or all solutions will be posted on our course page on bspace once homeworks are handed in.

- Homework 1 due on Friday, 8/31.
- Homework 2 due on Friday, 9/7.
- Homework 3 due on Friday, 9/14.
- Homework 4 due on Friday, 9/21.
- Homework 5 due on Friday, 9/28.
- Homework 6 due on Friday, 10/5.
- Homework 7 due on Friday, 10/12.
- No homework due on Friday, 10/19. Instead the take-home midterm is due that day.
- Homework 8 due on Friday, 10/26.
- Homework 9 due on Friday, 11/2.
- Homework 10 due on Friday, 11/9.
- Homework 11 due on Friday, 11/16.
- Homework 12a due on Friday, 11/30 (there will be a 12b, so it might be a good idea to complete 12a by 11/26 or so).
- Homework 12b due on Friday, 11/30 (along with 12a)

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