WEEK |
DATES |
TUTORIALS |
SECTIONS |
TOPICS |
1 |
Jan. 6-10 |
~ |
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 |
~ |
~ |
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, Z_{n}, 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 |
~ |
8.5, notes |
Constructing field of 256 elements. Efficient multiplication
and inversion. Advanced Encryption Standard. |