| Day | Topics | Preparation |
|---|---|---|
| Wed. 9/3 | Introduction. The Division Theorem. |
Sec. 1.1 |
| Fri. 9/5 | Proof of the division theorem. Properties of divisibility. |
Sec. 1.1-2 |
| Mon. 9/8 | The greatest common divisor and the Euclidean algorithm. | Sec. 1.2-3 |
| Wed. 9/10 | The GCD theorem and corollaries. Primes. | Sec. 1.2,3 |
| Fri. 9/12 | Uique factorization. Some exercises. |
Sec. 1.3 |
| Mon. 9/15 | Congruence modulo m in the integers. | Sec. 2.1-2 |
| Wed. 9/17 | Congruence class arithmetic, Z_m. | Sec. 2.2 |
| Fri. 9/19 | More congruence class arithmetic. | Sec. 2.2-3 |
| Mon. 9/22 | General rings and fields. | Sec. 3.1 |
| Wed. 9/24 | Questions before the exam? Product of rings. |
Sec. 3.1 |
| Fri. 9/26 | EXAMCh 1-2 | |
| Wed. 10/1 | Quiz Matrix rings. |
Sec. 3.1-2 |
| Fri. 10/3 | Matrix rings and Cartesian products of rings. Subrings. |
Sec. 3.1-2 |
| Mon. 10/6 | Properties of rings. Rings with identity. Idempotents |
Sec. 3.2 |
| Wed. 10/8 | Integral domains and fields.Sec 3.1-2 | |
| Fri. 10/10 | Homomorphism. Examples Z to Z_n Z_n to Z_d |
Sec. 3.3 |
| Mon. 10/13 | Homomorphisms and isomorphisms. | Sec. 3.3 |
| Wed. 10/15 | Isomorphisms. Non-isomorphic rings. | Sec. 3.3 |
| Fri. 10/17 | Polynomial arithmetic over rings. | Sec. 4.1 |
| Mon. 10/20 | Polynomial rings over a field. The division theorem. Divisibility |
Sec. 4.1-2 |
| Wed. 10/22 | The gcd and the Eulcidean algorithm. | Sec. 4.2 |
| Fri. 10/24 | Irreducible polynomials. | Sec. 4.2-3 |
| Mon. 10/27 | Unique factorization. | Sec. 4.3 |
| Wed. 10/29 | Roots and factors. | Sec. 4.4 |
| Fri. 10/31 | Roots and factors. | Sec. 4.4 |
| Mon. 11/3 | Problems. | Ch. 3-4 |
| Wed. 11/5 | Questions? Congruence in F[x]. |
Sec. 5.1 |
| Fri. 11/7 | EXAM. | Ch. 3, 4 |
| Mon. 11/10 | Congruence class arithmetic. | Sec. 5.1-2 |
| Wed. 11/12 | F[x]/m(x) for m(x) reducible and irreducible. | Sec. 5.2-3 |
| Fri. 11/14 | . | . |
| Mon. 11/17 | Ideals in rings. Examples. | Sec. 6.1 |
| Wed. 11/19 | Rings with only principal ideals.
New ideals from old: intersection, sum of ideals. |
Sec. 6.1 |
| Fri. 11/21 | A ring modulo an ideal. | Sec. 6.2 |
| Mon. 11/24 | Quotient rings and homomorphisms. | Sec. 6.2-3 |
| Wed. 11/26 | Some exercises: 5.3 #8,9, 6.1 #24, 6.2 #25. | Sec. 6.2 |
| Mon. 12/1 | Groups. Groups from rings. | Sec. 7.1 |
| Wed. 12/3 | First properties of groups. The order of an element. Matrix groups |
Sec. 7.1-2 |
| Fri. 12/5 | EXAM. | Ch. 5, 6. |
| Mon 12/8 | Permutation and symmetry groups. | Sec. 7.1-2 |
| Wed. 12/10 | Subgroups. | Sec. 7.3 |
| Fri. 12/12 | Homomorphisms and isomorphisms. | Sec. 7.4 |