Spring 2019
MATH 433–500: Applied Algebra
Time and venue: MWF 11:30 a.m.–12:20 p.m., BLOC 160
Office hours (BLOC 223b):
- MWF 10:15–11:15 a.m.
- by appointment
Additional office hours (BLOC 223b):
- Tuesday, April 30, 2:00–3:00 p.m.
- Wednesday, May 1, 10:15–11:15 a.m.
- Monday, May 6, 11:00 a.m.–2:00 p.m.
Final exam: Tuesday, May 7, 10:30 a.m.-12:30 p.m., BLOC 160
Rules for the exam: no books, no lecture notes, no computers. Bring paper and a stapler.
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]
Lecture 7: Invertible congruence classes.
- Humphreys/Prest 1.4 [exercises 3, 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 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: Functions. Relations.
- Humphreys/Prest 2.1 [exercises 6, 8], 2.2 [exercises 2(i-v), 3], 2.3 [exercises 1(a-f), 2(a-f)]
Lecture 15: Finite state machines.
- Humphreys/Prest 2.4 [exercises 1, 2, 4, 5]
Lecture 16: Permutations.
- Humphreys/Prest 4.1 [exercises 1, 2]
Lecture 17: Cycle decomposition. Order of a permutation.
- Humphreys/Prest 4.1 [exercises 3, 4(i-iii)], 4.2 [exercises 1(i-iv), 3, 4, 6, 7, 11]
Lecture 18: Sign of a permutation. Definition of the determinant.
- Humphreys/Prest 4.2 [exercises 1(i-iv), 10, 13]
Lecture 19: Alternating group. Abstract groups.
- Humphreys/Prest 4.2 [exercise 9], 4.3 [exercise 1(i-viii)]
Lecture 20: Abstract groups (continued).
- Humphreys/Prest 4.3 [exercises 1(i-viii), 2, 3, 4]
Lecture 21: Cayley table. Transformation groups.
- Humphreys/Prest 4.3 [exercises 5, 6, 7, 8]
Lecture 22: Semigroups.
- Humphreys/Prest 4.4 [exercises 1(i-v), 2]
Lecture 23: Rings. Fields.
- Humphreys/Prest 4.4 [exercises 3, 5, 6, 7]
Lecture 24: Rings and fields (continued).
- Humphreys/Prest 4.4 [exercises 4, 11, 12]
Lecture 25: Vector spaces over a field. Algebras over a field.
- Humphreys/Prest 4.4 [exercises 9, 10, 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, 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, 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 finite Abelian groups.
- Humphreys/Prest 5.3 [exercises 1(i-ii), 2, 3, 9]
Lecture 32: Error-detecting and error-correcting codes.
- Humphreys/Prest 5.4 [exercises 1, 2, 3]
Lecture 33: Linear codes (continued). Coset leaders and syndromes.
- Humphreys/Prest 5.4 [exercises 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 [exercise 1(i), 4]
Lecture 35: Greatest common divisor of polynomials. Factorisation of polynomials.
- Humphreys/Prest 6.2 [exercises 1(ii-iii), 2(i-iv), 3(i-ii)], 6.3 [exercises 2, 3, 4]
Lecture 36: Factorisation of polynomials (continued). Factorisation in general rings.
- Humphreys/Prest 6.3 [exercises 4, 5]
Lecture 37: Review for Exam 3.
- Humphreys/Prest 5.1-5.4, 6.1-6.3
Lecture 38: Factorisation in general rings (continued).
Lecture 39: Review for the final exam.
- Humphreys/Prest 1.1-1.6, 2.1-2.4, 4.1-4.4, 5.1-5.4, 6.1-6.3