| Day | Topics | Preparation |
|---|---|---|
| Mon. 8/31 | Introduction. The Division Theorem. |
Sec. 1.1 |
| Wed. 9/2 | The Euclidean algorithm. Properties of divisibility. |
Sec. 1.1-2 |
| Fri. 9/4 | The greatest common divisor theorem. Using the Euclidean algorithm to compute the GCD. | Sec. 1.3 |
| Wed. 9/9 | Primes and irreducibles. Unique Factorization. |
Sec. 3 |
| Fri. 9/11 | Proof of the unique factorization theorem.
Summary of Ch. 1. |
Sec. 1.1-3 |
| Mon. 9/14 | Congruence modulo m in the integers. | Sec. 2.1-2 |
| Wed. 9/16 | Congruence class arithmetic, Z_m. | Sec. 2.2 |
| Fri. 9/18 | More congruence class arithmetic. | Sec. 2.2-3 |
| Mon. 9/21 | General rings and fields. | Sec. 3.1 |
| Wed. 9/23 | Questions before the exam? Subrings. Properties of rings. |
Sec. 3.1-2 |
| Fri. 9/25 | EXAMCh 1-2 | |
| Mon. 9/28 | Cartesian product of rings. Matrix rings. |
Sec. 3.1-2 |
| Wed. 9/30 | Z[sqrt(2)] Matrix rings, the quaternions. | Sec. 3.2 |
| Fri. 10/2 | Integral domains and fields.Sec 3.1-2 | |
| Mon. 10/5 | Homomorphisms. Examples Z to Z_n Z_n to Z_d |
Sec. 3.3 |
| Wed. 10/7 | Homomorphisms and isomorphisms. | Sec. 3.3 |
| Fri. 10/9 | Examples and problems on isomorphisms. | Sec. 3.3 |
| Mon. 10/12 | Polynomial arithmetic over rings. | Sec. 4.1 |
| Wed. 10/14 | Polynomial rings over a field. The division theorem. Divisibility |
Sec. 4.1-2 |
| Fri. 10/16 | The gcd and the Euclidean algorithm. | Sec. 4.2 |
| Mon. 10/19 | Irreducible polynomials. | Sec. 4.2-3 |
| Wed. 10/21 | Unique factorization. | Sec. 4.3 |
| Fri. 10/23 | Roots and factors. | Sec. 4.4 |
| Mon. 10/26 | Roots and factors. | Sec. 4.5 |
| Wed. 10/28 | Problems. | Ch. 3-4 |
| Fri. 10/30 | Congruence in F[x]. | Sec. 5.1 |
| Mon. 11/2 | Congruence class arithmetic. | Sec. 5.1-2 |
| Wed. 11/4 | EXAM. | Ch. 3, 4 |
| Fri. 11/6 | F[x]/m(x) for m(x) reducible and irreducible. | Sec. 5.2-3 |
| Mon. 11/9 | . | Sec. 5.2-3 |
| Wed. 11/11 | Holiday | . |
| Fri. 11/13 | Ideals in rings. Examples. | Sec. 6.1 |
| Mon. 11/16 | Rings with only principal ideals.
New ideals from old: intersection, sum, product, radical. |
Sec. 6.1 |
| Wed. 11/18 | Two rings with non-principal ideals A ring modulo an ideal. |
Sec. 6.2 |
| Fri. 11/20 | Quotient rings and homomorphisms. | Sec. 6.2-3 |
| Mon. 11/23 | Some exercises: 5.3 #8,9, 6.1 #24, 6.2 #25. | Sec. 6.2 |
| Wed. 11/25 | FURLOUGH DAY: NO CLASS. | . |
| Mon. 11/30 | Groups. Groups from rings. | Sec. 7.1 |
| Wed. 12/2 | First properties of groups. The order of an element. Matrix groups |
Sec. 7.1-2 |
| Fri. 12/4 | EXAM. | Ch. 5, 6. |
| Mon 12/7 | Permutation and symmetry groups. | Sec. 7.1-2 |
| Wed. 12/9 | Subgroups. | Sec. 7.3 |
| Fri. 12/11 | Homomorphisms and isomorphisms of groups. | Sec. 7.4 |