## Algebra — Math 30710 — Spring 2019 |

**Course Information** -- info-Math30710.pdf

The text book is "A first course in Abstract Algebra, 7th edition", by John B. Fraleigh, Addison Wesley, 2003, ISBN 0-201-76390-7.Your instructor is Prof. Katrina Barron; Office 276C Hurley; Phone: 574-631-3981; E-mail: kbarron@nd.edu.

**Resources**

Prof. K. Barron's office hours are:

MWF after class in DBRT 209 and then in DBRT 207 if there is still ongoing discussion after 3:30pm;

and Tuesday 4:00-5:00 in Hurley 276C;

or by appointment.Math Bunker Help: There will be a pair of undergraduate and/or graduate student TAs in the Math Bunker (the room in the southeast corner of the math library) Sunday through Thursday between 7:00 and 9:00pm. The aim is to help with homework, exam prep, etc. in regular and honors math courses that emphasize rigor and abstraction (i.e. proofs) rather than primarily computation. You can go to ask specific questions, to hear other people's questions or just to work in a friendly supportive environment. No appointment necessary. Please contact Katie Gallagher (kgalla17@nd.edu) if you have any questions.

Tutoring available: The Mathematics Library offers free drop-in tutoring/help sessions for any math class during Sunday through Thursday evenings from 7pm to 11pm. More information on this opportunity, how to sign up for a session or to see availability for a drop-in, visit the Mathematics Library website at Math Library Tutoring .

**Course Updates and Announcements**

The Final Exam is Tuesday, May 7th, from 4:15–6:15 in our usual classroom, DBRT 209. The exam will be comprehensive, i.e. covering material from all homework assignments, but will focus a bit more on the material not covered by the three midterm exams, i.e. HW 12 & 13, Sections 18, 19, 20, 22, 23, & 24. For those with exam conflicts on Tuesday, the exam will also be given on Wednesday, May 8th, from 10am to Noon in Hayes-Healy 229.

**Schedule and Homework** — Homework is to be turned in in class on the day indicated below on which it is due. All the pages of your assignment must be stapled together.

__Schedule and Homework Assignments__

Date |
Reading & Topics |
Assignment |
Due Date |

W 1/16 | Section 0, pp. 1–8, Intro, Sets, Partitions, & Residue classes modulo n |
Section 0, pp. 8–10: #3, 5, 7, 11, 25, 28, 30, 31, 34, 35, 36. | HW 1: W 1/23 |

F 1/18 | Sections 1 & 2, pp. 11–25, Equivalence Relations, Complex numbers, Modular Arithmetic & Binary operations |
Section 1, p. 19: #3, 7, 11, 12, 13, 16, 19, 20, 22, 26, 30, 32, 34, 36, 38, 40. Section 2, pp. 26–28: #5–8, 10, 12, 14, 16, 23, 36, 37. |
HW 1: W 1/23 |

M 1/21 | More on Sections 1 & 2; and Section 3, pp. 14–33, Complex Numbers & Isomorphic Binary Structures. |
Section 3, pp. 34–36: #2, 3, 4, 6, 7, 10, 16, 26–28, 32, 33. | HW 2: M 2/4 |

W 1/23 | Sections 3 & 4, pp. 30–45, Isomorphic Binary structures & Intro to Groups |
Section 4, pp. 45–46: #4–8, 10, 14, 16–18, 20. | HW 2: M 2/4 |

F 1/25 | Sections 3& 4, pp. 30–45, Structural Properties of Binary Operations & Groups + in class quiz |
Section 4, pp. 46–49: #22–24, 29–32, 34–36, 40, 41. | HW 2: M 2/4 |

M 1/28 | Sections 3 & 4 pp. 31–45, More on Structural Properties & Groups |
Continue to work on homework assigned for Sections 3 & 4. | None. |

W 1/30 | Class canceled! | Continue to work on homework assigned for Sections 3 & 4. Bring questions to class on Friday. | None. |

F 2/1 | Questions and examples on Sections 3 & 4 Section 5, pp. 49–55, Subgroups |
Section 5, pp. 55–59: #4, 6–8, 12, 20, 23, 24, 26, 30–32, 36, 41, 43–45, 51, 52, 57. | HW 3: F 2/8 |

M 2/4 | Sections 5 & 6, pp. 51–65, Subgroups & Cyclic Groups |
Section 6, pp. 66–68: #8, 10, 14, 15, 18, 20, 21, 22, 25, 28. | HW 4: F 2/15 |

W 2/6 | Sections 5 & 6, pp. 53–65, More on Cyclic Groups |
Section 6, pp. 66–68: #30, 31, 32, 36, 37, 38, 40, 44, 49, 50, 55. | HW 4: F 2/15 |

F 2/8 | Section 6, pp. 61–65 Cyclic Groups |
Continue to work on Section 6 homework assigned above. | HW 4: F 2/15 |

M 2/11 | Section 6, pp. 63–65 Classification of Cyclic Groups and Subgroups + in class quiz on Sections 3 & 4. |
Continue to work on Section 6 homework assigned above. | None. |

W 2/13 | Sections 6 & 7, pp. 64 & 69–70 & dihedral-quaternions.pdf Generating Sets & Dihedral Groups. |
Section 7, pp. 72–73: #2–5, 19. | HW 5: F 2/22 |

F 2/15 | Sections 7 & 8, pp. 69–70 & 75–83, Dihedral & Quaternions & Groups of Permutations |
Please do the extra exercises here: dihedral-quaternion-homework.pdf. | HW 5: F 2/22 |

M 2/18 | Review for Exam I | Please bring your questions on the material for Midterm Exam I — Sections 0–5, Homeworks 1–3. | None. |

T 2/19 | Exam I | Midterm Exam I will cover Sections 0–5, Homeworks 1–3. You may take the exam either from 3:30–5:00 in DBRT 209; or from 6:30–8:00 in Hayes-Healy 125. | None. |

W 2/20 | Section 8, pp. 75–83, Dihedral & Quaternions & Groups of Permutations |
Section 8, pp. 83–84: #2–6, 8, 9, 10. | HW 6: W 2/27 |

F 2/22 | Section 8, pp. 75–83, Groups of Permutations & Cayley’s Theorem |
Section 8, pp. 84–86: #11, 17, 18, 21, 24, 30, 31, 36, 46, 47. | HW 6: W 2/27 |

M 2/25 | Section 8, pp. 81–83, Cayley’s Theorem |
Please do the extra exercises here: hw7.pdf. | HW 7: F 3/8 |

W 2/27 | Sections 9, pp. 87–93, Orbits & Cycles |
Section 9, pp. 94–96: #2, 6, 7, 10–13, 14, 17, 29, 34, 35. Also do #23 but do not hand in. |
HW 7: F 3/8 |

TH 2/28 | Extra Credit Opportunities | Math 4 Everyone Lecture,
5pm Jordan Hall 101, or Delahanty Math Series, 5:30pm Hayes-Healy 127. Turn in questionnaire that will be available soon via email by Monday 3/4. |
Extra: M 3/4. |

F 3/1 | Sections 9 & 10, pp. 90–101, Transpositions, the Alternating Group, Cosets & Lagrange’s Theorem + Short in class quiz |
Section 10, pp. 101–102: #1–4, 12, 15, 16, 20–24. Also do #19 but do not hand in. |
HW 7: F 3/8 |

M 3/4 | Sections 9 & 10, pp. 91–101, Even v. Odd Permutations & Cosets & Lagrange’s Theorem |
Continue to work on HW 7. | None. |

W 3/6 | Sections 10 & 11, pp. 100–110, Lagrange’s Theorem & Direct Products |
Section 10, pp. 102–104: #26–32, 34, 36, 37, 39, 46, 47. | HW 8: M 3/25 |

F 3/8 | Section 11, pp. 108–110, Direct Products & The Classification of Finitely Generated Abelian Groups |
Section 11, pp. 110–112: #1, 2, 4, 6—9, 13, 14, 16, 18, 20, 22, 24, 26, 32. | HW 8: M 3/25 |

MWF 3/11-15 | None. | SPRING BREAK!!! | None. |

M 3/18 | Sections 11, pp. 108–109 & hw9.pdf, Elementary Divisors v. Invariant Factors |
Section 11, pp. 112–113: #36, 46, 47, 49. Also do the extra exercises here: hw9.pdf |
HW 9: F 3/29 |

W 3/20 | Review for Exam II | Bring questions to class on material for Midterm Exam II: Sections 6–10, Homework Assignments 4–7, including extra problems from pdf files. | None. |

TH 3/21 | Exam II | Midterm Exam II covers Sections 6–10, Homework Assignments 4–7, including extra problems from pdf files. | None. |

F 3/22 | Sections 11 & 13, pp. 109–133 & hw9.pdf, Internal v. External Direct Products, & Homomorphisms |
Continue to work on the pdf for Homework 9. | HW 9: F 3/29 |

M 3/25 | Sections 13 & 14, pp. 135–141, Properties of Homomorphisms |
Section 13, pp. 133–134: #3, 4, 5, 18, 20, 22, 24—29. | HW 9: F 3/29 |

W 3/27 | Sections 13 & 14, pp. 130–141, Normal Groups, Factor Groups, & The Fundamental Homomorphism Theorem. |
Section 13, pp. 133–134: #32, 33, 34, 36, 39, 42, 44, 45, 47, 49, 51, 52. | HW 10: W 4/3 |

F 3/29 | Sections 14, pp. 138–141, Normal Groups, Factor Groups, & The Fundamental Homomorphism Theorem. |
Section 14, pp. 141–143: #1, 2, 6, 8, 9, 10, 12, 14, 17, 18, 19, 23a—d, g—j. | HW 10: W 4/3 |

M 4/1 | Sections 14 & 15, pp. 138–151, Normal Groups, Factor Groups, & The Fundamental Homomorphism Theorem. |
Section 14, pp. 141–143: #27, 28, 29, 31—37, 40. | HW 11: W 4/10 |

W 4/3 | Sections 14 & 15, pp. 144–151, Factor Groups |
Section 15, pp. 151–154: #1–5, 6, 10, 13, 14, 19, 28—31, 34, 35, 37, 38, 42. | HW 11: W 4/10 |

F 4/5 | Sections 14 & 15, pp. 144–151, Factor Groups, Commutators, Centers, & Simple Groups + in class quiz on Sections 10 & 11. Intro to Rings and Fields, Section 18 pp. 167—174. |
Read Section 18. Section 18, pp. 174–175: #2, 4, 6, 7, 8, 9, 11, 12, 13. |
HW 11: W 4/10 |

M 4/8 | Section 18 pp. 167—174. Rings, Fields, & Ring Homomorphisms |
Section 18, pp. 175—177: #14, 16, 17—20, 22—25, 27, 28, 32, 33, 37—42, 50, 52. | HW 12: W 4/24 |

W 4/10 | Sections 19 & 24, pp. 177—182, & pp. 220-226. Integral Domains, Fields, & Noncommutative Rings |
Section 19, pp. 182—183: #1–4, 6, 8, 10, 12, 14, 15, 17—20, 29. Section 24, pp. 226—227: #2, 4, 6, 11, 14. |
HW 12: W 4/24 |

F 4/12 | Sections 19 & 24, pp. 177—182, & pp. 220-226. Integral Domains, Fields, & Noncommutative Rings + in class quiz |
Please do the extra exercises here: hw12.pdf | HW 12: W 4/24 |

M 4/15 | Review for Exam III | Bring questions from HW 8—11, Sections 10, 11, 13, 14, 15. | None. |

T 4/16 | Exam III | Exam III will cover HW 8—11, Sections 10, 11, 13, 14, 15, including extra problems from the pdf file for HW 9. | None. |

W 4/17 | Section 20, pp. 184—189, Fermat’s and Euler’s Theorems & Section 22, pp. 198—207, Polynomial Rings |
Section 20: pp. 189—190: #2, 4, 6, 8, 10, 12—18, 20, 22, 23, 24—28. Read Section 22. |
HW 13: W 5/1 |

F-M 4/19-4/22 | Easter Break! | None. | None. |

W 4/24 | Finish Section 20, Euler’s Theorem Read Section 22, pp. 198—207, Polynomial Rings |
Section 22, pp. 207—208: #1, 2, 4, 6, 8, 10, 12, 14, 16, 17, 22—25. | HW 13: W 5/1 |

TH 4/25 | Extra Credit Opportunity | Math 4 Everyone Lecture, 5pm Jordan Hall 101 | None. |

F 4/26 | Sections 22 & 23, pp. 198—218, Factorization of Polynomials and Applications |
Section 23, pp. 218—219: #2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 28, 30. Also do the extra exercises here: hw13.pdf |
HW 13: W 5/1 |

M 4/29 | Section 23, pp. 213—218, Eisenstein’s Criteria, Extension Fields & Intro to Galois Theory. |
Continue to work on assignments for HW 13. | None. |

W 5/1 | Groups, Fields & Applications | In class work on Galois Extensions, Finite Fields, and Classification of Finite Simple Groups. | None. |

TH-T 5/2-5/7 | Extra Office Hours |
There will be extra office hours as follows, but please also feel free to make an appointment or provide feedback as to when would be best for these office hours: Thursday 5/2: 2:00-3:00pm Friday 5/3: 2:00-3:00pm Monday 5/6: 2:00-3:00pm Tuesday 5/7: 2:00-3:00pm |
None. |

T 5/7 | Final Exam | The Final Exam is 4:15-6:15pm in our usual room DBRT 209. The exam will also be given on Wednesday 10am—Noon in Hayes-Healy 229 to accommodate those with exam conflicts. | None. |

Last updated May 2, 2019.