Math 113, Section 1, Summer 2020

Information for students

Syllabus
Zoom link (class and office hours): see syllabus, or use meeting ID: 967 1902 4950, password: 12257
bCourses Site
Piazza site
Gradescope site
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

There is no required text for this course; any of several algebra textbooks will be great. The course has been listed as requiring Fraleigh's A first course in abstract algebra or Artin's Algebra (2^nd edition), but you may also find these useful: Abstract Algebra by Dummit and Foote and Abstract Algebra by Thomas Judson (the latter is free online here). We will plan to cover material in chapters 2, 6, 7, 11–15 of Artin, all of which is contained in every textbook. Relevant readings for Artin, Judson, and Fraleigh will be indicated below.

Homework, Readings, etc.

(tentative; will be updated throughout the course)

June 22: Intro to the course; sets, set operations, relations, functions, binary operations, equivalence relations
   Reading: Artin 2.7, Judson Chapter 1, Fraleigh 0, 2
   Problems: Judson, Chapter 1: 6, 8, 23, 28; Fraleigh, Section 0: 1, 3, 12, 26, 32, 33
June 23: Modular arithmetic, groups, examples of groups
   Reading: Artin 2.1–2.2, 2.7, 2.9, Judson 3.1–3.2, 4.2, 12.1, Fraleigh 1–2, 4
   Problems: Artin, Chapter 2: 1.1, 2.1, 2.3b; Judson, Chapter 2: 1f, 2, 10, 28
June 24: Subgroups, subgroups of (ℤ, +), cyclic subgroups
   Reading: Artin 2.2–2.4, Judson 2.1, 3.3, 4.1, Fraleigh 5–6
   Problems: Artin, Chapter 2: 2.4, 3.2, 4.3, 4.5, 4.7, 4.9, 4.10; Fraleigh, Chapter 5: 20, 23, 26, 27, 55; Fraleigh, Chapter 6: 30, 33, 48, 56
June 25: Homomorphisms, isomorphisms
   Reading: Artin 2.5, Judson 9.1, 11.1, Fraleigh 3, 13
   Problems: Artin, Chapter 2: 5.1, 5.5, 6.2, 6.4, 6.7, 6.10a, 6.10b
Homework 1 (due on July 1): Click here

June 29: Deep dive on certain groups: integers mod n, symmetric groups, dihedral groups
   Reading: Artin 1.5, 2.9, 6.4, Judson Chapter 5, Fraleigh 8–9
   Problems: Artin, Chapter 1: 5.1, 5.4; Artin, Chapter 2: 9.1; Artin, Chapter 6: 4.1; Judson, Chapter 5: 2, 5, 10, 19, 34
June 30: Cosets, normal subgroups, (start of) quotient groups
   Reading: Artin 2.8, 2.12, Judson Chapter 6, 10.1, Fraleigh 10, 13–14
   Problems: Artin, Chapter 2:  8.1, 8.3, 8.4, 8.6, 8.9; Fraleigh, Section 10: 1, 4, 12, 19, 34, 45, 46
July 1: Quotient groups, First Isomorphism Theorem
   Reading: Artin 2.12, Judson 11.1, Fraleigh 14–15, 34
   Problems: Artin, Chapter 2: 12.1, 12.4; Judson, Chapter 10: 2, 4, 11; Fraleigh, Section 14: 1, 9, 31; Fraleigh, Section 34: 1, 7
July 2: Product groups, symmetries, isometries
   Reading: Artin 2.11, 6.1–6.2, Judson 9.2, 12.2, Fraleigh 11–12
   Problems: Artin, Chapter 2: 11.4abc, 11.8; Artin, Chapter 6: 3.3, 3.6, 4.3; Judson, Chapter 9: 23, 48; Fraleigh, Section 11: 15–18, 46, 50; Fraleigh, Section 12: 1
Homework 2 (due on July 8): Click here

July 6: Isometries of ℝⁿ, orthogonal groups, quaternions
   Reading: Artin 6.2–6.3, Judson 12.2, Fraleigh 12, Stillwell 7.6, 7.8 (need to be on UCB network/proxy)
   Problems: Judson, Chapter 12: 10, 13; see July 2 Problems for more
July 7: Group actions, orbit–stabiliser theorem
   Reading: Artin 6.7–6.9, Judson 14.1, Fraleigh 16–17
   Problems: Artin, Chapter 6: 7.1, 7.2, 7.10 (b means the coefficients are in ℤ₅), 8.2, 9.1; Judson, Chapter 14: 2, 3, 5, 20
July 8: Conjugation action, class equation, permutation representation, Cayley's theorem
   Reading: Artin 6.10–6.11, 7.1–7.3, Judson 9.1, 14.2, Fraleigh 8, 37
   Problems: Artin, Chapter 6: 11.1, 11.3; Artin, Chapter 7: 2.3, 2.7, 2.14, 3.3; Fraleigh, Section 37: 1, 8–10
July 9: Sylow theorems
   Reading: Artin 7.6–7.7, Judson Chapter 15, Fraleigh 36
   Problems: Artin, Chapter 7: 7.3, 7.5, 7.9, 8.4; Judson, Chapter 15: 7, 12, 16
Homework 3 (due on July 13): Click here

July 13: Generating sets, group presentations, classification of finitely generated Abelian groups
   Reading: Artin 7.9–7.10, Fraleigh 39, 40, Wikipedia on free groups and presentations
   Problems: Artin, Chapter 7: 9.2, 10.3, 10.4, 10.5; Fraleigh, Section 39: 1, 3, 4
July 14: Bonus topics, some of: semi-direct products, simple groups, colouring knots with algebra, cryptography, Rubik's cube
July 15: Review day
July 16: Midterm
Homework 4 (due on July 22): Click here

July 20: Rings, examples
   Reading: Artin 11.1–11.2, Judson 16.1, 17.1, Fraleigh 18, 22–23
   Problems: Artin, Chapter 11: 1.3, 1.6, 1.7, 2.1; Judson, Chapter 16: 1, 3, 30, 31, 34; Judson, Chapter 17: 2, 3
July 21: Ring homomorphisms, ideals
   Reading: Artin 11.3, Judson 16.2, Fraleigh 26
   Problems: Artin, Chapter 11: 3.2, 3.3, 3.9, 3.12; Judson, Chapter 16: 6, 7, 25, 27, 36; Fraleigh, Section 26: 30
July 22: Quotient rings
   Reading: Artin 11.4, Judson 16.2, Fraleigh 26
   Problems: Artin, Chapter 11: 4.3, 4.4; Fraleigh, Section 26: 3, 10
July 23: Product rings, adjoining elements, fractions
   Reading: Artin 11.5–11.7, 13.5, Judson 16.2, 18.1, Fraleigh 19, 21
   Problems: Artin, Chapter 11: 5.1, 5.2, 5.3, 5.4, 5.7, 6.1, 6.2 (just ℤ/(8)), 7.1, 7.2; Judson, Chapter 16: 38; Fraleigh, Section 19: 29; Fraleigh, Section 21: 1, 2, 4, 12; Fraleigh, Section 26: 15
Homework 5 (due on July 29): Click here

July 27: Maximal/prime ideals, primes and irreducibles
   Reading: Artin 11.8, 12.1, 13.5, Judson 16.4, Fraleigh 27
   Problems: Artin, Chapter 11: 8.1, 8.3, Artin, Chapter 12: 2.1, 2.4, 4.4, Judson, Chapter 16: 4
July 28: Euclidean domains, principal ideal domains, unique factorisation domains
   Reading: Artin 12.2, Judson 17.3, 18.2, Fraleigh 45–47
   Problems: Artin, Chapter 12: 2.6b, 2.7, Judson, Chapter 17: 3, 6, 9, 14, Fraleigh, Section 46: 1, 5, 17, 18
July 29: PID vs ED, UFD vs PID, ℤ[𝑥] is a UFD
   Reading: Artin 12.3, Judson 17.2–17.3, Fraleigh 45
   Problems: Artin, Chapter 12: 3.2, Fraleigh, Section 45: 9, 10, 14, 24, 28, 30
July 31: Factoring integer polynomials, Gaussian primes
   Reading: Artin 12.4–12.5, Fraleigh 47
   Problems: Artin, Chapter 12: 4.5, 4.9, 4.12, 5.1, 5.3
Homework 6 (due on August 5): Click here

August 3: Field extensions, algebraic/transcendental elements, degrees of field extensions
   Reading: Artin 15.1–15.4, Judson Chapter 20, 21.1–21.2, Fraleigh 29–31
   Problems: Artin, Chapter 15: 1.1, 2.1, 2.2, 3.6, 3.9, M.1
August 4: More on degrees of field extensions, algebraic extensions, compass/straightedge constructibility
   Reading: Artin 15.4–15.5, Judson 21.1, 21.3, Fraleigh 31–32
   Problems: Artin, Chapter 15: 3.1, 3.9, 3.10, Fraleigh, Section 31: 3, 10, 31
August 5: More compass/straightedge constructibility
   Reading: Artin 15.5, Judson 21.3, Fraleigh 32
   Video: Construction of 17-gon (other videos: 1 and 2)
August 6: Finite fields, simple and separable extensions
   Reading: Artin 15.7–15.8, Judson 22.1, Fraleigh 33
   Problems: Artin, Chapter 15: 3.5, 6.3, 7.3, 7.7
Homework 7 (due on August 10): Click here

August 10: Tour of Galois theory
August 11: Tour of Galois theory
August 12: Review day
August 13: Final exam

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