MATH 433, section 100

Summer 2010
MATH 433–100: Applied Algebra

Time and venue: MTWRF 12:00–1:35 p.m., BLOC 160

First day handout

Office hours:

TWRF 10:30–11:30 a.m.

by appointment

Additional office hours:

Monday, July 5, 1–3 p.m.

Course schedule:

Part I: Number theory and more

Mathematical induction

Euclidean algorithm

Primes, factorisation

Congruence classes, modular arithmetic

Euler's theorem, public key encryption

Functions, relations

Finite state machines

Humphreys/Prest: Chapters 1–2

Lecture 1: Greatest common divisor. Euclidean algorithm.

Humphreys/Prest 1.1 [exercises 1(i), 1(ii), 1(iii), 1(iv), 2, 4, 7]

Lecture 2: Mathematical induction. Prime numbers. Unique factorisation theorem.

Humphreys/Prest 1.2 [exercises 1, 2, 8, 12], 1.3 [exercises 1, 2, 3(a), 8]

Lecture 3: Prime factorisation (continued). Congruence classes. Modular arithmetic.

Humphreys/Prest 1.1 [exercises 5, 6], 1.3 [exercises 3(b), 5], 1.4 [exercises 1(i), 1(ii), 1(iii), 2]

Lecture 4: Modular arithmetic (continued). Linear congruences.

Humphreys/Prest 1.4 [exercises 3(i), 3(ii), 3(iii), 5, 9(ii), 9(iii)], 1.5 [exercise 1(i–vii)]

Lecture 5: Chinese remainder theorem. Fermat's little theorem. Euler's theorem.

Humphreys/Prest 1.5 [exercises 2(i), 2(ii), 3, 5], 1.6 [exercises 1(i–iv), 2(i–iv), 7]

Lecture 6: Euler's totient function. Public key systems.

Humphreys/Prest 1.6 [exercises 5, 6(i–iii), 12]

Lecture 7: 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 8: Review for Exam 1.

Humphreys/Prest 1.1–1.6, 2.3

Lecture 9: Finite state machines.

Humphreys/Prest 2.4 [exercises 1, 2, 4, 5]

Part II: Abstract algebra

Permutations

Abstract group theory

Cosets, Lagrange's theorem

Other algebraic structures (rings, fields, etc.)

Error-correcting codes

Humphreys/Prest: Chapters 4–5

Lecture 10: Permutations.

Humphreys/Prest 4.1 [exercises 1, 2, 3, 4], 4.2 [exercises 3, 6, 7]

Lecture 11: Order and sign of a permutation.

Humphreys/Prest 4.2 [exercises 1(i–iv), 2, 11(i–iii)]

Lecture 12: Sign of a permutation (continued). Abstract groups.

Humphreys/Prest 4.2 [exercises 9, 10, 13], 4.3 [exercises 1(i–viii), 2]

Lecture 13: Examples of groups.

Humphreys/Prest 4.3 [exercises 3, 4, 5]

Lecture 14: Further examples of groups. Semigroups.

Humphreys/Prest 4.3 [exercises 7, 8], 4.4 [exercises 1(i–v)]

Lecture 15: Rings. Fields. Vector spaces over a field.

Humphreys/Prest 4.4 [exercises 3(i–iii), 3(v), 3(viii), 4, 6, 9(i–iii)]

Lecture 16: Algebraic structures (continued).

Humphreys/Prest 4.4 [exercises 10(i–ii), 10(iv), 11, 12, 13]

Lecture 17: Order of an element in a group. Subgroups.

Humphreys/Prest 5.1 [exercises 1, 2, 4(i–iii), 5]

Lecture 18: Cyclic groups. Cosets. Lagrange's theorem.

Humphreys/Prest 5.1 [exercises 3, 10], 5.2 [exercises 1, 3, 5]

Lecture 19: Subgroups (continued). Error-detecting and error-correcting codes.

Humphreys/Prest 5.2 [exercise 5], 5.4 [exercise 1]

Lecture 20: Binary codes. Linear codes.

Humphreys/Prest 5.4 [exercises 2, 4]

Lecture 21: Linear codes (continued). Classification of groups.

Humphreys/Prest 5.3 [exercises 1(i–ii), 4, 8], 5.4 [exercises 6, 7]

Lecture 22: Review for Exam 2.

Humphreys/Prest 4.1–4.4, 5.1–5.4

Summer 2010 MATH 433–100: Applied Algebra