Math 113, Fall 2019

Sections 4 and 8

Information for students

Syllabus
bCourses Site
Piazza site
GSI: Kentarô Yamamoto (office hours details at the link)
DSP students should speak to the instructor as soon as possible, even if you don't have a letter yet.
Guidelines on what to do if you think you may have a conflict between this class and your extracurricular activities. In particular, you must speak to the instructor before the end of the second week of classes.
Academic honesty in mathematics courses: A statement on cheating and plagiarism, courtesy of Michael Hutchings.
How to get an A in this class, courtesy of Kathryn Mann.

Textbook

The required text for this course is Algebra (2^nd edition) by Michael Artin. Other helpful texts are A first course in abstract algebra by Fraleigh, Abstract Algebra by Dummit and Foote, and Abstract Algebra by Thomas Judson. From Artin, we plan to cover material in chapters 2, 6, 7, 11–15.

Homework, Readings, etc.

(will be updated throughout the course)

August 28: Sets, set operations, relations, functions, binary operations
   Reading: Section 2.7 of Artin, Chapter 1 of Judson
August 30: More on relations, equivalence relations, modular arithmetic, intro to groups
   Reading: Sections 2.7, 2.9 of Artin, Chapter 1 of Judson
   Reading: Section 2.1 of Artin
September 2: No class! Labour day!
September 4: Groups, examples of groups, subgroups
   Reading: Sections 2.1–2.2 of Artin
 * Homework 1 (due Wednesday, September 11): Artin Chapter 2, problems 1.1, 2.1, 2.3b, 2.4, 4.3, 4.5, 4.7, 4.9, 4.10
September 6: Subgroups of (Z, +), cyclic subgroups
   Reading: Sections 2.3–2.4 of Artin
September 9: Cyclic subgroups, homomorphisms
   Reading: Sections 2.4–2.5 of Artin
September 11: Homomorphisms, isomorphisms
   Reading: Sections 2.5–2.6 of Artin
 * Homework 2 (due Wednesday, September 18; turn in only the starred questions): Artin Chapter 2, problems *5.1, 5.5, 6.1, *6.2, 6.3, 6.4, 6.7, *6.10a, 6.10b, *9.7
September 13: Group of integers mod n, symmetric group
   Reading: Sections 1.5, 2.9 of Artin, Section 5.1 of Judson
September 16: Cosets, normal subgroups
   Reading: Section 2.8 of Artin
September 18: Quotient groups, First Isomorphism Theorem
   Reading: Section 2.12 of Artin
 * Homework 3 (due Wednesday, September 25; turn in only the starred questions): Artin Chapter 2, problems 8.1, *8.4, *8.6, 8.9, 9.4, 11.4ab, *11.4c, 11.8, 12.1, *12.4, 12.5
September 20: Product groups
   Reading: Section 2.11 of Artin
September 23: Symmetries, isometries
   Reading: Sections 6.1–6.2 of Artin
September 25: Isometries of R^n, orthogonal groups
   Reading: Sections 6.2–6.3 of Artin
September 27: Group actions
   Reading: Sections 6.7–6.8 of Artin
September 30: Midterm 1 (in class)
   Closed book (ie. no notes, textbook, or any other material allowed)
   Material: Everything we covered in class up to and including September 23
      Most of the material of Artin, Chapter 2 (except 2.10) and 6.1–6.2 (modulo the material we didn't cover in class)
   Preparation: past midterms are on bCourses
   Preparation: try Artin Chapter 2, problems 2.2, 2.6, 4.1, 4.2, 4.6, 5.4, 6.9, 7.1, 8.3, 8.5, 9.1, 11.5, 12.2, M.2, M.3, M.9, M.10, and Chapter 6 (last page of PDF), problems 4.2
October 2: Orbit–Stabiliser theorem
   Reading: Sections 6.8–6.9 and 7.3 of Artin
 * Homework 4 (due Wednesday, October 9; turn in only the starred questions): Artin Chapter 6, problems *3.3, 3.6, 4.1, 7.1, *7.2, 7.3, *7.10 (b means the coefficients are integers mod 5), 8.2, *9.1
October 5: Conjugation action; class equation
   Readng: Sections 7.2–7.3 of Artin
October 7: Group actions on subsets, permutation representations, Cayley's theorem
   Reading: Sections 6.10–6.11 and 7.1 of Artin
October 9: No class; campus closed
October 11: No class; campus closed
October 14: Group presentations
   Reading: Sections 7.9–7.10 of Artin
October 16: Rings
   Reading: Sections 11.1–11.2 of Artin
 * Homework 5 (due Wednesday, October 23; turn in only the starred questions): Artin Chapter 6 (page 1 of PDF): 11.1, *11.3, 11.6 (F₃ = Z₃); Chapter 7 (page 2–3 of PDF): 2.3, *2.7, 2.8, *2.14, *3.3, 3.4
October 18: Polynomial rings, ring homomorphisms
   Reading: Sections 11.2–11.3 of Artin
October 21: Ring homomorphisms, ideals
   Reading: Section 11.3 of Artin
October 23: Ideals, quotient rings
   Reading: Sections 11.3–11.4 of Artin
 * Homework 6 (due Wednesday, October 30; turn in only the starred questions): Artin, Chapter 11, problems *1.3, 1.6, 1.7, 2.1, 3.2, *3.3, 3.9a, *3.12, 4.3, *4.4
October 25: Quotient rings
   Reading: Section 11.4 of Artin
October 28: No class; campus closed
October 30: Quotient rings, product rings
   Reading: Sections 11.4, 11.6 of Artin
 * Homework 7 (due Wednesday, November 6; turn in only the starred questions): Artin, Chapter 11, problems *5.1, 5.2, *5.3, 5.4, 5.7, 6.1, *6.2 (just Z/(8)), 6.5, *7.1, 7.2
November 1: Adjoining elements, fractions
   Reading: Sections 11.5, 11.7 of Artin
November 4: Primes and irreducibles
   Reading: Sections 12.1–12.2 of Artin
Noveber 6: Euclidean domains, principal ideal domains
   Reading: Section 12.2 of Artin
November 8: Unique factorisation domains
   Reading: Section 12.2 of Artin
November 13: PID vs ED, UFD vs PID
   Reading: Section 12.3 of Artin
 * Homework 8 (due Wednesday, November 20; turn in only the starred questions: Artin, Chapter 12, problems *2.1, 2.2, *2.4, *2.6b, 2.7, *3.2, 3.6, 4.4
November 15: Midterm 2 (in class)
   Closed book (ie. no notes, textbook, or any other material allowed)
   Material: Artin sections 6.7–6.9, 6.11, 7.1–7.3, 11.1–11.7 (not 11.4.2a nor 11.4.3 nor the term faithful), 12.2 (up to the end of page 360)
   Preparation: past midterms/questions are on bCourses
   Preparation: try Artin Chapter 6: 7.6, 7.7, 8.1, 9.4, 11.2; Chapter 7: 1.1, 2.2, 2.9bcd, 2.13, 2.17; Chapter 11: 1.8, 1.9, 3.1, 3.6, 3.11, 3.13, 4.3, 5.5, 5.6, 6.4, 7.3; Chapter 12: 2.1, 2.2, 2.4
November 18: Z[x] is a UFD
   Reading: Section 12.3 of Artin
November 20: Factoring integer polynomials
   Reading: Section 12.4 of Artin
November 22: Fields, algebraic/transcendental elements
   Reading: Section 15.1–15.2 of Artin
November 25: Field extensions by algebraic elements
   Reading: Section 15.2 of Artin
 * Homework 9 (due Wednesday, December 4; turn in only the starred questions: Artin, Chapter 15, problems *1.1, 2.1, *2.2, *3.6, 3.9, *3.10
December 2: Degree of a field extension
   Reading: Section 15.3 of Artin
December 4: Compass and straightedge constructibility
   Reading: Section 15.5 of Artin
December 6: Compass and straightedge constructibility
   Reading: Section 15.5 of Artin
December 16: Final exam (in our classroom)
   Closed book (ie. no notes, textbook, or any other material allowed)
   Sections covered: 2.1–2.9, 2.11–2.12, 6.4 (dihedral group), 6.7–6.9, 6.11, 7.1–7.3, 11.1–11.7, 12.1–12.4, 15.1–15.3, 15.5
      Whatever material in those sections that we didn't cover in class is not testable!
   Additional helpful sections: 1.5 (permutations), 3.2–3.4 (helpful for vector spaces, bases), 11.8 (maximal ideals), 15.4 (finding minimal polynomials)
   Additional practice questions from Chapter 1: 5.1
   Additional practice questions from Chapter 3: 2.8a, 2.11, 4.3, 4.8, M.5
   Additional practice questions from Chapter 11: 8.1, 8.3
   Additional practice questions from Chapter 12: 4.1, 4.2, 4.5, 4.12, M.7
   Additional practice questions from Chapter 15: 3.1, 4.2b, 5.2a, M.1, M.4d (multiplicative group F\{0} of a field F)

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