# Math 31: Abstract Algebra

## syllabus

date
description
Sept 10 why abstract algebra? five motivations: historical, higher mathematics, applications to other scientific disciplines, abstraction of the familiar, because it's there Herstein sections 1.2, 1.3, and Scott LaLonde's set theory handout
Sept 11 (x-hour) proof writing workshop part 1: logic, proof by contradiction
Sept 12 some number theory basics; modular arithmetic: two points of view Herstein p. 23-26 (Euclidean algorithm), p. 40-41, and section 2.2
Sept 14 modular arithmetic, definition of group
Sept 17 examples of groups, definitions, first theorems Herstein section 2.1 and 2.3
Sept 18 (x-hour) proof writing workshop part 2: techniques, examples, style
Sept 19 the mattress group, cyclic groups, subgroups Fraleigh section 5 and Herstein p. 61-63
Sept 21 the integers mod n: addition and multiplication Herstein section 3.1
Sept 24 integers mod n under multiplication, dihedral groups
Sept 26 equivalence relations, symmetric groups Herstein section 2.4
Sept 28 cosets and Lagrange's theorem
Oct 1 cosets, Lagrange, homomorphism Herstein section 2.5
Oct 2 (x-hour) symmetry group of the tetrahedron
Oct 3 tetrahedron, permutations, cycle decomposition, homomorphism yoga
Oct 5 a counterexample to the converse of Lagrange, homomorphisms, isomorphisms Herstein section 2.6
Oct 8 isomorphism, quaternions
Oct 10 quotient groups, quaternions
Oct 12 isomorphism theorem, quaternions Herstein theorem 2.6.4, p. 96-98
Oct 15 Cauchy's theorem, direct products
Oct 16 (x-hour) midterm study session bring questions
Oct 17 direct products, structure theorem for finite abelian groups
Oct 19 in class portion of midterm Herstein section 4.1
Oct 22 applications of group theory to elliptic curves, rings
Oct 24 rings: examples and basic definitions Herstein 4.2, 4.3
Oct 26 fields, oddtown, quadratic rings
Oct 29 ring homomorphisms, ideals Herstein 4.4
Oct 30 (x-hour) optional: midterm debrief
Oct 31 halloween, quotient rings, maximal ideals Herstein 5.1
Nov 2 ideals, quadratic rings Herstein 4.6
Nov 5 polynomial rings tomorrow is election day: please go vote
Nov 7 irreducibles, polynomials Herstein 5.3
Nov 9 roots of polynomials, field extensions Herstein 5.4, 5.5
Nov 12 field extensions, constructibility Herstein 5.6
Nov 13 (x-hour) a bit of galois theory enjoy your take home final

worksheets