Algebraic Structures with Computer Applications

MATH3101, Fall 2007

WEEK

DATES

TUTORIALS

SECTIONS

TOPICS

~

Sep. 6-7

~

1.1
Introduction to the course. Sets.

1

Sep. 10-14

~

1.1-1.4, 1.6
Sets. Binary operations. Maps and composition. Equivalence relations.

2

Sep. 17-21

Review of first week with examples.

Notes
Monoids. Automata and formal languages.

3

Sep. 24-28

Induction proofs and group excercises.

3.1, 3.2
Groups definitions and examples. Order of a group.

4

Oct. 1-5

Test 1 (monoids and groups)

3.2-3.3, 3.3
Subgroups. Exponents. Cyclic subgroups.

5

Oct. 8-12

Division algorithm and Euclidean algorithm on integers and ploynomials.

3.4, 3.5
Cyclic groups. Generators. Infinite and finite cyclic groups. Isomorphisms.

6

Oct. 15-19

Test 2 (cyclic groups, homorphisms)

4.1, 4.2
Homomorphisms. Kernel and image. Permutation groups.

7

Oct. 22-26

Review of groups.

4.4 and notes, 4.5
Cycle notation. Transpositions. Conjugates.

8

Oct. 19-Nov.2

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

5.1, 5.2
Cosets. Lagrange's theorem. Normal subgroups. Applications to public key cryptography.

9

Nov. 5-9

Review of Rings.

5.3,6.1
RSA. Normal subgroups. Quotient groups.

10

Nov. 12-16

Test 4 (rings and fields)

6.7 and notes, 6.4
Homomorphism theorem. Rings and subrings. Inverses.

11

Nov. 19-23

Polynomial factorization, rings and fields.

8.1, 8.2

Fields. Ideals. Quotient rings. Maximal ideals.

12

Nov. 26-30

Examples of ideals and fields.

8.3 and 8.5 and notes
Constructing a field of 256 elements. Efficient multiplication and inversion. Advanced Encryption Standard.

The above weekly outline is subject to change depending on the progress of the course.
Last modified: August 26, 2007

WEEK	DATES	TUTORIALS	SECTIONS	TOPICS
~	Sep. 6-7	~	1.1	Introduction to the course. Sets.
1	Sep. 10-14	~	1.1-1.4, 1.6	Sets. Binary operations. Maps and composition. Equivalence relations.
2	Sep. 17-21	Review of first week with examples.	Notes	Monoids. Automata and formal languages.
3	Sep. 24-28	Induction proofs and group excercises.	3.1, 3.2	Groups definitions and examples. Order of a group.
4	Oct. 1-5	Test 1 (monoids and groups)	3.2-3.3, 3.3	Subgroups. Exponents. Cyclic subgroups.
5	Oct. 8-12	Division algorithm and Euclidean algorithm on integers and ploynomials.	3.4, 3.5	Cyclic groups. Generators. Infinite and finite cyclic groups. Isomorphisms.
6	Oct. 15-19	Test 2 (cyclic groups, homorphisms)	4.1, 4.2	Homomorphisms. Kernel and image. Permutation groups.
7	Oct. 22-26	Review of groups.	4.4 and notes, 4.5	Cycle notation. Transpositions. Conjugates.
8	Oct. 19-Nov.2	Test 3(permutation groups, Lagrange's theorem, Homomorphism theorem)	5.1, 5.2	Cosets. Lagrange's theorem. Normal subgroups. Applications to public key cryptography.
9	Nov. 5-9	Review of Rings.	5.3,6.1	RSA. Normal subgroups. Quotient groups.
10	Nov. 12-16	Test 4 (rings and fields)	6.7 and notes, 6.4	Homomorphism theorem. Rings and subrings. Inverses.
11	Nov. 19-23	Polynomial factorization, rings and fields.	8.1, 8.2	Fields. Ideals. Quotient rings. Maximal ideals.
12	Nov. 26-30	Examples of ideals and fields.	8.3 and 8.5 and notes	Constructing a field of 256 elements. Efficient multiplication and inversion. Advanced Encryption Standard.