Zoom link (class and office hours): see syllabus, or use meeting ID: 967 1902 4950, password: 12257

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Piazza site

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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.

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, 33June 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, 28June 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, 56June 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, 6.10bHomework 1 (due on July 1):Click here

June 29:Deep dive on certain groups: integers modn, 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, 34June 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, 46July 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, 7July 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: 1Homework 2 (due on July 8):Click here

July 6:Isometries of ℝ^{n}, 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 moreJuly 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 ℤ_{5}), 8.2, 9.1; Judson, Chapter 14: 2, 3, 5, 20July 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, 8–10July 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, 16Homework 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, 4July 14:Bonus topics, some of: semi-direct products, simple groups, colouring knots with algebra, cryptography, Rubik's cubeJuly 15:Review dayJuly 16:MidtermHomework 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, 3July 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: 30July 22:Quotient rings Reading: Artin 11.4, Judson 16.2, Fraleigh 26 Problems: Artin, Chapter 11: 4.3, 4.4; Fraleigh, Section 26: 3, 10July 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: 15Homework 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: 4July 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, 18July 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, 30July 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, 5.3Homework 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.1August 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, 31August 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, 7.7Homework 7 (due on August 10):Click here

August 10:Tour of Galois theoryAugust 11:Tour of Galois theoryAugust 12:Review dayAugust 13:Final exam

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