| Day | Topics | Preparation |
|---|---|---|
| Wed. 1/23 | Introduction. The Division Theorem. |
Sec. 1.1 |
| Fri. 1/25 | Uniqueness in the division theorem. Divisors of an integer. |
Sec. 1.1-2 |
| Mon. 1/28 | Divisibility and the
Euclidean algorithm. More on the gcd. | Sec. 1.2-3 |
| Wed. 1/30 | Primes and unique factorization. | Sec. 1.3 |
| Fri. 2/1 | Primes and unique factorization. | Sec. 1.3 |
| Mon. 2/4 | Congruence modulo m in the integers. | Sec. 2.1-2 |
| Wed. 2/6 | Congruence class arithmetic, Z_m. | Sec. 2.2 |
| Fri. 2/8 | More congruence class arithmetic. | Sec. 2.2-3 |
| Mon. 2/11 | General rings and fields. | Sec. 3.1 |
| Wed. 2/13 | Questions before the exam? Product of rings. Matrix rings |
Sec. 3.1 |
| Fri. 2/15 | EXAMCh 1-2 | |
| Mon. 2/18 | Properties of rings. | Sec. 3.2 |
| Wed. 2/20 | Rings with identity. Integral domains. Examples,Z[sqrt(2)] , Z[1/2] . |
Sec. 3.2 |
| Fri. 2/22 | Integral domains and fields.Ch 3.1-2 | |
| Mon. 2/25 | Homomorphism. Examples Z to Z_n Homomorphisms Z_n to Z_d |
Sec. 3.3 |
| Wed. 2/27 | Homomorphisms and isomorphisms. | Sec. 3.3 |
| Fri. 2/29 | Problem: The characteristic of a ring; Non-isomorphic rings. |
Ch. 3 |
| Mon. 3/3 | Polynomial arithmetic over rings. | Sec. 4.1 |
| Wed. 3/5 | The ring of continuous functions. The characteristic of a ring; Non-isomorphic rings. |
Ch. 3 |
| Fri. 3/7 | The division theorem. | Sec. 4.1 |
| Mon. 3/10 | Divisibility , the gcd. | Sec. 4.2 |
| Wed. 3/12 | The Euclidean algorithm. Irreducible polynomials. |
Sec. 4.2-3 |
| Fri. 3/14 | Unique factorization. | Sec. 4.3 |
| Mon. 3/17 | Roots and factors. | Sec. 4.4 |
| Wed. 3/19 | Roots and factors. | Sec. 4.4 |
| Fri. 3/21 | Roots and factors. Congruence in F[x]. |
Sec. 4.4 Sec. 5.1 |
| Mon. 3/24 | Congruence class arithmetic. | Sec. 5.1-2 |
| Wed. 3/26 | Universal properties of polynomial rings. Questions on Ch. 3,4? | Ch. 3, 4 |
| Fri. 3/28 | EXAM. | Ch. 3, 4 |
| Mon. 4/7 | Arithmetic of F[x]/p(x) | Sec. 5.2-3 |
| Wed. 4/9 | F[x]/p(x) for p(x) reducible and irreducible. | Sec. 5.2-3 |
| Fri. 4/11 | Retake EXAM. | Ch. 3, 4 |
| Mon. 4/14 | Ideals in rings. Examples. | Sec. 6.1 |
| Wed. 4/16 | Rings with only principal ideals.
New ideals from old: intersection, sum of ideals. |
Sec. 6.1 |
| Fri. 4/18 | A ring modulo an ideal. | Sec. 6.2 |
| Mon. 4/21 | Quotient rings and homomorphisms. | Sec. 6.2-3 |
| Wed. 4/23 | Some exercises: 5.3 #8,9, 6.1 #24, 6.2 #25. | Sec. 6.2 |
| Fri. 4/25 | EXAM. | Ch. 5, 6. |
| Mon. 4/28 | Groups. Groups from rings. | Sec. 7.1 |
| Wed. 4/30 | First properties of groups. The order of an element. Matrix groups |
Sec. 7.1-2 |
| Fri. 5/2 | Permutation and symmetry groups. | Sec. 7.1-2 |
| Mon 5/5 | Subgroups. | Sec. 7.3 |
| Wed. 5/7 | Homomorphisms and isomorphisms. | Sec. 7.4 |