Spring 2024
MATH 433–500: Applied Algebra
Time and venue: MWF 1:50–2:40 p.m., BLOC 160
Office hours (BLOC 301b):
- MWF 11:00 a.m.–12:00 p.m.
- or by appointment
Office hours (ZOOM meeting):
- Thursday 5:00–6:00 p.m.
- or by appointment
Quiz 1: Monday, January 22 (topics: greatest common divisor, Euclidean algorithm)
Quiz 2: Friday, January 26 (topics: mathematical induction, prime factorisation)
Quiz 3: Friday, February 2 (topics: congruences, modular arithmetic)
Quiz 4: Friday, February 9 (topics: linear congruences, Chinese Remainder Theorem)
Exam 1: Friday, February 16
Quiz 5: Monday, February 26 (topics: relations, finite state machines)
Quiz 6: Friday, March 1 (topics: permutations, cycle decomposition, order of a permutation)
Quiz 7: Friday, March 8 (topic: abstract groups)
Quiz 8: Friday, March 22 (topics: semigroups, rings and fields)
Exam 2: Wednesday, March 27
Quiz 9: Friday, April 5 (topics: order of an element in a group, subgroups, cyclic groups)
Quiz 10: Friday, April 12 (topics: Lagrange's Theorem, classification of finite Abelian groups)
Quiz 11: Friday, April 19 (topics: the ISBN code, error-detecting and error-correcting binary codes)
Exam 3: Friday, April 26
Quiz 12: Monday, April 29 (topics: division of polynomials, factorisation of polynomials)
Course schedule:
Part I: Number theory
- Mathematical induction
- Euclidean algorithm
- Primes, factorisation
- Congruence classes, modular arithmetic
- Euler's theorem
- Public key encryption
Humphreys/Prest: Chapter 1
Lecture 1: Division of integers. Greatest common divisor.
- Humphreys/Prest 1.1 [exercises 1(i), 1(ii)]
Lecture 2: Euclidean algorithm.
- Humphreys/Prest 1.1 [exercises 1(iii), 1(iv), 2, 7]
Lecture 3: Mathematical induction.
- Humphreys/Prest 1.2 [exercises 1, 2, 8, 12]
Lecture 4: More on greatest common divisor. Prime numbers. Unique factorisation theorem.
- Humphreys/Prest 1.1 [exercises 4, 5], 1.3 [exercises 1, 2, 3(a-b), 5, 8]
Lecture 5: Prime factorisation (continued). Congruences.
- Humphreys/Prest 1.3 [exercises 6, 7, 9], 1.4 [exercise 1(i-vi)]
Lecture 6: Congruences (continued). Modular arithmetic.
- Humphreys/Prest 1.4 [exercises 2, 5, 9(i-ii)]
Lecture 7: Invertible congruence classes.
- Humphreys/Prest 1.4 [exercises 3(i-v), 4, 6]
Lecture 8: Linear congruences.
- Humphreys/Prest 1.5 [exercise 1(i-vii)]
Lecture 9: Chinese Remainder Theorem.
- Humphreys/Prest 1.5 [exercises 2(i-iii), 3, 5]
Lecture 10: Order of a congruence class. Fermat's Little Theorem.
- Humphreys/Prest 1.6 [exercises 1(i-iv), 2(i-iv), 3, 4, 7]
Lecture 11: Euler's Theorem. Euler's phi-function.
- Humphreys/Prest 1.6 [exercises 5, 6(i-iii), 8]
Lecture 12: Public key encryption. The RSA system.
- Humphreys/Prest 1.6 [exercises 9, 12, 13]
Lecture 13: Review for Exam 1.
Part II: Abstract algebra and more
- Functions, relations
- Finite state machines
- Permutations
- Abstract groups
- Other algebraic structures (rings, fields, etc.)
Humphreys/Prest: Chapters 2 and 4
Lecture 14: Sets and functions. Relations.
- Humphreys/Prest 2.1 [exercises 4, 6, 8], 2.2 [exercises 2(i-v), 5(i-iii), 6]
Lecture 15: Relations (continued). Finite state machines.
- Humphreys/Prest 2.3 [exercises 1(a-g), 2(a-f), 3], 2.4 [exercises 1(a-b), 2(a-c)]
Lecture 16: Finite state machines (continued). Permutations.
- Humphreys/Prest 2.4 [exercises 3(a-d), 4, 5(i-iii)]
Lecture 17: Permutations (continued). Cycle decomposition.
- Humphreys/Prest 4.1 [exercises 1, 2]
Lecture 18: Cycle decomposition (continued). Order of a permutation.
- Humphreys/Prest 4.1 [exercises 3, 4(i-iii)], 4.2 [exercises 1(i-iv), 3, 7, 11(i-iii)]
Lecture 19: Order and sign of a permutation. Alternating group.
- Humphreys/Prest 4.2 [exercises 1(i-iv), 9, 13]
Lecture 20: Sign of a permutation (continued). Classical definition of the determinant.
- Humphreys/Prest 4.2 [exercises 10, 13]
Lecture 21: Abstract groups.
- Humphreys/Prest 4.3 [exercises 1(i-viii), 2, 3, 4]
Lecture 22: Basic properties of groups. Cayley table. Transformation groups.
- Humphreys/Prest 4.3 [exercises 3, 5, 6, 8]
Lecture 23: Semigroups.
- Humphreys/Prest 4.4 [exercises 1(i-v), 2]
Lecture 24: Rings and fields.
- Humphreys/Prest 4.4 [exercises 3(i-viii), 4(i-iii), 5, 6, 12, 13]
Lecture 25: Rings and fields (continued). Vector spaces over a field.
- Humphreys/Prest 4.4 [exercises 9(i-iii), 11, 13]
Lecture 26: Review for Exam 2.
- Humphreys/Prest 2.1-2.4, 4.1-4.4
Part III: Group theory and polynomials
- Subgroups, cyclic groups
- Cosets, Lagrange's theorem
- Classification of groups
- Error-detecting and error-correcting codes
- Division of polynomials
- Factorisation of polynomials
Humphreys/Prest: Chapters 5–6
Lecture 27: Properties of groups. Order of an element in a group.
- Humphreys/Prest 5.1 [exercises 1, 2, 5, 8, 9]
Lecture 28: Subgroups. Cyclic groups.
- Humphreys/Prest 5.1 [exercises 4(i-iv), 6, 10]
Lecture 29: Cosets. Lagrange's Theorem.
- Humphreys/Prest 5.2 [exercises 1, 2, 3, 5]
Lecture 30: Direct product of groups. Quotient group.
- Humphreys/Prest 5.3 [exercises 4, 5, 6, 8]
Lecture 31: Isomorphism of groups. Classification of groups.
- Humphreys/Prest 5.3 [exercises 1(i-iii), 3, 9]
Lecture 32: Error-detecting and error-correcting codes.
- Humphreys/Prest 5.4 [exercises 1, 3]
Lecture 33: Linear codes. Coset leaders and syndromes.
- Humphreys/Prest 5.4 [exercises 2, 4, 5, 6]
Lecture 34: Polynomials in one variable. Division of polynomials.
- Humphreys/Prest 6.1 [exercises 1(i-vi), 2(i-vi)], 6.2 [exercises 1(i), 4]
Lecture 35: Zeros of polynomials (continued). Greatest common divisor of polynomials.
- Humphreys/Prest 6.1 [exercise 3(i-iii)], 6.2 [exercise 1(ii-iii)]
Lecture 36: Euclidean algorithm for polynomials. Factorisation of polynomials.
- Humphreys/Prest 6.2 [exercises 2(i-v), 3(i-iii)], 6.3 [exercises 2, 3, 4, 5]
Lecture 37: Review for Exam 3. (preliminary lecture notes)
- Humphreys/Prest 5.1-5.4, 6.1-6.3