Spring 2015
  • MATH 433–500: Applied Algebra
  • Time and venue:   MWF 11:30 a.m.–12:20 p.m., BLOC 160

    First day handout


    Office hours (BLOC 223b):

    Quiz 1:  Monday, January 26 (topics: greatest common divisor, Euclidean algorithm)
    Quiz 2:  Friday, January 30 (topics: mathematical induction, prime factorisation)
    Quiz 3:  Friday, February 6 (topics: congruences, modular arithmetic)
    Quiz 4:  Friday, February 13 (topics: linear congruences, Chinese Remainder Theorem)
    Exam 1:  Friday, February 20
    Quiz 5:  Monday, March 2 (topics: relations, finite state machines)
    Quiz 6:  Friday, March 6 (topic: permutations)
    Quiz 7:  Friday, March 13 (topic: abstract groups)
    Quiz 8:  Friday, March 27 (topics: semigroups, rings, fields, vector spaces over a field)
    Exam 2:  Wednesday, April 1
    Quiz 9:  Friday, April 10 (topics: order of an element in a group, cyclic groups)
    Quiz 10:  Friday, April 17 (topics: Lagrange's Theorem, isomorphism of groups)
    Quiz 11:  Friday, April 24 (topics: the ISBN code, error-detecting and error-correcting binary codes)
    Exam 3:  Friday, May 1
    Quiz 12:  Monday, May 4 (topics: division of polynomials, factorisation of polynomials)
    Final exam:  Tuesday, May 12

    Sample problems for Exam 1

    Sample problems for Exam 2

    Sample problems for Exam 3



    Course schedule:

    Part I: Number theory


    Humphreys/Prest: Chapter 1


    Lecture 1: Division of integers. Greatest common divisor.
    Lecture 2: Euclidean algorithm.
    Lecture 3: Mathematical induction.
    Lecture 4: More on greatest common divisor. Prime numbers. Unique factorisation theorem.
    Lecture 5: Prime factorisation (continued). Congruences.
    Lecture 6: Congruences (continued). Congruence classes.
    Lecture 7: Modular arithmetic. Invertible congruence classes.
    Lecture 8: Linear congruences.
    Lecture 9: Chinese Remainder Theorem.
    Lecture 10: Order of a congruence class. Fermat's Little Theorem.
    Lecture 11: Euler's phi-function.
    Lecture 12: Public key encryption. The RSA system.
    Lecture 13: Review for Exam 1.

    Part II: Abstract algebra and more


    Humphreys/Prest: Chapters 2 and 4


    Lecture 14: Functions. Relations.
    Lecture 15: Finite state machines.
    Lecture 16: Permutations.
    Lecture 17: Cycle decomposition. Order of a permutation.
    Lecture 18: Sign of a permutation.
    Lecture 19: Alternating group. Abstract groups.
    Lecture 20: Abstract groups (continued).
    Lecture 21: Transformation groups.
    Lecture 22: Semigroups. Rings.
    Lecture 23: Fields. Vector spaces over a field.
    Lecture 24: More on algebraic structures.
    Lecture 25: Review for Exam 2.

    Part III: Group theory and polynomials


    Humphreys/Prest: Chapters 5–6


    Lecture 26: Order of an element in a group. Subgroups.
    Lecture 27: Subgroups (continued). Cyclic groups.
    Lecture 28: Cyclic groups (continued). Cosets. Lagrange's Theorem.
    Lecture 29: Lagrange's Theorem (continued). Classification of subgroups. Quotient group.
    Lecture 30: Isomorphism of groups. Classification of Abelian groups.
    Lecture 31: Isomorphism of groups (continued). The ISBN code.
    Lecture 32: Error-detecting and error-correcting codes. Binary codes. Linear codes.
    Lecture 33: Linear codes (continued). Coset leaders and syndromes.
    Lecture 34: Polynomials in one variable. Division of polynomials.
    Lecture 35: Greatest common divisor of polynomials. Factorisation of polynomials.
    Lecture 36: Review for Exam 3.