Algebraic Structures with Computer Applications

MATH3101, Winter 2003

MATH3101, CLASS OUTLINE FOR WINTER 2003

WEEK 

DATES 

TUTORIALS 

SECTIONS 

TOPICS 

1

Jan. 6-10

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1.1-1.2, 1.3-1.4 and 1.6

Sets, binary operations, maps and composition, equivalence relations.

2

Jan. 13-17

Review of first week with examples.

Notes, 3.1

Monoids, automota and formal languages. Groups definitions and examples.

3

Jan. 20-24

Induction proofs and group excercises.

3.1, 3.2

Examples of groups, order of a group, subgroups, exponents. Cyclic subgroups.

4

Jan. 27-31

Test 1 (monoids and groups)

3.3, 3.3

Cyclic groups, generators, infinite and finite cyclic groups, order of elements.

5

Feb. 3-7

Division algorithm and Euclidean algorithm on integers and polynomials.

3.4, 3.5

Isomorphisms, homomorphisms, kernel, image.

6  

Feb 10-14

Test 2 (cyclic groups, homorphisms)

4.1, 4.4

Permutation groups and cycle notation. Cosets. Lagrange's theorem.

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Feb. 17-21

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Reading Week

7

Feb 24-28

Review of groups.

4.4 and notes, 4.5

Lagrange's theorem. Primality testing and public key cryptography. Normal subgroups. Quotient groups.

8

Mar. 3-7

Test 3 (permutation groups, Lagrange's theorem, Homomorphism theorem)

4.5-5.1, 5.2

Homomorphism theorem. Rings, subrings, inverses, zero divisors, integral domains.

9

Mar. 10-14

Review of rings.

5.2, 5.3

Fields, Zn, field of fractions. Ring isomorphisms.

10

Mar. 17-21

Test 4 (rings and fields)

6.1 and notes, 8.1

Ideals, ring cosets, quotient ring. Fields from rings. Examples of fields. Ring polynomials.

11

Mar. 24-28

Complex numbers.

8.2, 8.3


Extended Euclidean algorithm, greatest comon divisor of polynomials. Polynomial factorization.

12 

Mar 31-Apr. 4  

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8.5, notes

Constructing field of 256 elements. Efficient multiplication and inversion. Advanced Encryption Standard.