Math H113, Spring 2020

Information for students

Syllabus
bCourses Site
Piazza site
GSI: Benjamin Castle, 961 Evans, Mon 1–3, Tue 2–5, Wed 12–2, Thu 12–3
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, 9, 11–16 (although mileage may vary).

Homework, Readings, etc.

(will be updated throughout the course)

January 21: Introduction to group theory
   Reading: Know material in Section 2.7 of Artin, Chapter 1 of Judson
   Reading (today's material): Artin Sections 2.1–2.2, Fraleigh, pages 41–42
January 23: Examples of groups, subgroups
   Reading: Artin Sections 1.5, 2.2–2.3, 2.9, Judson Section 5.1
January 28: primes in ℤ, cyclic (sub)groups
   Reading: Artin Sections 2.3–2.4
 * Homework 1 (due on Tuesday, February 4): click here
January 30: homomorphisms and isomorphisms, cosets
   Reading: Artin Sections 2.5–2.6 and 2.8
February 4: cosets, normal subgroups, quotient groups
   Reading: Artin Sections 2.8, 2.12
 * Homework 2 (due on Tuesday, February 11): click here
February 6: first isomorphism theorem, (semi)direct products
   Reading: Artin Sections 2.11–2.12
   Reading: Dummit and Foote, pages 175–180 on semidirect product
February 11: group extensions, dihedral groups, group actions
   Reading: Wikipedia page on group extensions
   Reading: Artin Sections 6.4 (dihedral groups), 6.7
 * Homework 3 (due on Tuesday, February 18): click here
February 13: group actions
   Reading: Artin Sections 6.7–6.11, 7.1
February 18: class equation, p–groups
   Reading: Artin Sections 7.2–7.3, 7.5
 * Homework 4 (due on Tuesday, February 25): click here
February 20: Sylow theorems
   Reading: Artin Sections 7.6–7.7
February 25: solvable and simple groups
   Reading: Artin Section 7.4
   Reading: Dummit and Foote, pages 105, 194–196 on solvable groups
   Reading: Proof that A_n is simple for n≥5
   Recommended (but optional) reading: Artin Sections 7.9–7.10 on group presentations
February 27: rings
   Reading: Artin Sections 11.1–11.2
March 3: homomorphisms, ideals
   Reading: Artin Sections 11.3
 * Homework 5 (due on Thursday, March 12): click here
March 5: Midterm (in class)
   Material: everything up to the end of Feb 25 class, all homework questions
             but not: semigroups; monoids; things covered in the textbook that we didn't talk about in class
   Closed book (ie. no notes, textbook, or any other material allowed)
   Practice problems on material not covered by the homework: Artin Section 7: 4.7, 4.8, 4.9, 5.2, 5.12, 7.3, 7.4, 7.9, 8.4, 8.5
        Prove that every p–group is solvable; that a p–group has a normal subgroup of every possible index; that if |G| = p^em, then G has a subgroup of order pⁱ for each i = 1, ..., e
   I will not ask you to reprove anything we did in class, but you should understand the techniques that go into the proofs, and how to use them
   You might need to give definitions of terms from class
March 10: no class!
March 12: quotient rings, product rings
   Video: on bCourses; 5:10 quotient rings, 23:10 example 1, 26:30 example 2, 43:30 example 3, 1:03:20 product rings, 1:06:50 some category theory (optional), 1:20:40 final questions
   Reading: Artin Sections 11.4, 11.6
March 17: product rings, idempotents, adjoining elements, fractions, primes and irreducibles
   Video: on bCourses
   Reading: Artin Sections 11.6–11.7, 12.2
 * Homework 6 (due on Tuesday, March 31): click here
March 19: primes and irreducibles, Euclidean domains, principal ideal domains, unique factorisation domains
   Video: on bCourses
   Reading: Artin Section 12.2
March 31: unique factorisation domains, Z[x] is a UFD, factoring in Z[x]
   Video: on bCourses
   Reading: Artin Sections 12.2, 12.4
 * Homework 7 (due on Tuesday, April 7): click here
April 2: fields, algebraic/transcendetal elements, degree of extensions, splitting fields
   Reading: Artin Sections 15.1–15.4, 15.6
April 7: finite fields, more on splitting fields, Galois theory introduction
   Reading: Artin Sections 15.7, 16.3–16.4
 * Homework 8 (due on Tuesday, April 14): click here
April 9: characteristic 0 fields and primitive elements, fixed fields, splitting fields = Galois extensions
   Reading: Artin Sections 15.8, 16.5–16.6
April 14: fundamental theorem of Galois theory, solvability by radicals
   Reading: Artin Section 16.7, Judson Section 23.3
 * Homework 9 (due on Tuesday, April 21): click here
April 21: solvability by radicals (converse), symmetric functions, solving cubics/quartics
   Reading: Artin Sections 16.8–16.8, Stewart Sections 18.4–18.5, Lemma 10.1
 * Homework 10 (due on Tuesday, April 28): click here
April 28: straightedge and compass constructibility, regular n–gons
   Reading: Artin Section 15.5, Judson Section 21.3, and notes here

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