# Math 31: Topics in Algebra

SYLLABUS

This is a tentative course syllabus and it will be updated irregularly. The page will be updated every lecture..

 Lectures Sections in Text Brief Description Day 1:  1/5 Chapter 0 Properties of integers; Euclidean algorithm, Modular arithmetic Day 2:  1/7 Chapters 0 and 1 Mathematical Induction, Equivalence Relations, Properties of Functions, First Examples of groups Day 3:  1/8 x-hour instead of  Saturday 1/10 Chapter 2 Binary operations, Definition and further examples of groups. Day 4:  1/9 Chapter 2, and 3 Elementary properties of groups, Finite groups, Subgroups. Day 5:  1/12 Chapter 3 Subgroups, Subgroup tests. Day 6:  1/14 Chapter 3 and start Chapter 4 Center of a group, Centralizer of an element. Cyclic groups: definitions and examples 1/15 optional x-hour problem solving session and preparation for the quiz Day 7:  1/16 Chapter 4 Properties of cyclic groups and start classification of subgroups of cyclic groups. 1/19 Martin Luther King Jr. Day No class Day 8:  1/21 1st 30 minute quiz Chapter 4 Classification of subgroups of cyclic groups. Day 9:  1/22 x-hour instead of the class on 1/19 Chapter 5 Permutations; Cycle notation and Properties Day 10:  1/23 Chapter 5 Properties of permutations Day 11:  1/26 Chapter 6 Group isomorphism; Examples Day 12:  1/28 Chapter 6 Properties of an isomorphism; Automorphisms 1/29 optional x-hour problem solving session and preparation for the Midterm exam Day 13:  1/30 Chapter 7 Cosets; Lagrange's theorem; Arithmetical corollaries Day 14:  2/2 Midterm exam Chapter 7 Fermat's little theorem; Orbits and stabilizers; The group of rotations of the cube Day 15:  2/4 Chapter 8 External Direct Product of Groups Day 16:  2/6 Chapter 9 Normal subgroups, Factor Groups, Hölder's theorem Day 17:  2/9 Chapter 9 More comments on normal subgroups, Application of Factor Groups, Internal Direct Product Day 18:  2/11 Chapter 10 Group Homomorphisms, properties of subgroups under homormorphisms Day 19:  2/12 x-hour instead of the class on 2/13 Chapter 10 Normal subgroups and kernels, First Isomorphism Theorem 2/13 Winter Carnival No class J Day 20:  2/16 The date of 2nd 30 minute quiz is changed to 2/18 following students wishes Chapters 10 Applications and Examples Day 21:  2/18 2nd 30 minute quiz Chapters 12 Rings and examples of rings Day 22:  2/20 Chapters 12 and 13 Subrings, and Integral Domains Day 23:  2/23 Chapter 13 Fields, Characteristic of a Ring with Unity Day 24:  2/25 Chapter 14 Ideals and Factor Rings, prime and maximal ideals 2/26 optional x-hour problem solving and preparation for the quiz Day 25:  2/27 Chapter 15 Ring Homomorphisms, properties, ideals are kernels Day 26:  3/1 3rd 30 minute quiz Chapter 15 Finish Ring Homomorphisms Day 27:  3/3 Chapter 16 Polynomial rings, division of polynomials Day 28:  3/5 Chapter 17 Factorization of Polynomials, Irreducible polynomials Day 29:  3/8 Chapter 19, 20, 21 Vector spaces and their dimensions (briefly). Extensions and algebraic extensions of fields.