Abstract Algebra: Math 521A, Fall 2010
Schedule



Text: Hungerford, Abstract Algebra : An Introduction

Schedule

My best approximation.

Matrix rings
Questions before the exam? Subrings of matrix rings.
The quaternions.
Cartesian product of rings. Integral domains and fields.
Examples: Z[1/2] ,Z[sqrt(2)] Homomorphisms.
Examples Z to Z_n Z_n to Z_d
Day Topics Preparation
Mon. 8/30 Introduction. Sec. 1.1
Wed. 9/1 The Division Theorem.
Properties of divisibility.
Sec. 1.1-2
Fri. 9/3 The Euclidean algorithm.
The greatest common divisor theorem.
Sec. 1.2
Wed. 9/8 Primes and irreducibles.
Unique Factorization.
Sec. 1.3
Fri. 9/10 GCD and unique factorization exercises.
Congruence modulo n
Sec. 1.1-3
Mon. 9/13 Congruence modulo m in the integers.
Congruence class arithmetic, Z_m.
Sec. 2.1-2
Wed. 9/15 Sage: meet in 425.
Z_n in Sage.
Sec. 2.2
Fri. 9/17 Units and zero divisors in Z_n. Sec. 2.2-3
Mon. 9/20 Solving equations in Z_n.
Using the Euclidean algorithm to compute the GCD.
Sec. 2.2-3
Wed. 9/22 General rings and fields.
Subrings. Properties of rings.

Sec. 3.1-2
Fri. 9/24
Sec. 3.1-2
Mon. 9/27 EXAM Ch 1-2
Wed. 9/29 Sage: meet in 425.
Euclidean algorithm, etc.
Sec. 1.1, 3.1-3
Fri. 10/1 Subrings of Z and Z_n. Sec.3.1-2
Mon. 10/4 Sec.3.1-2
Wed. 10/6 Sec 3.1-2
Fri. 10/8 Sec. 3.3
Mon. 10/11 Homomorphisms and isomorphisms. Sec. 3.3
Wed. 10/13 Isomorphisms and onon-isomorphic rings
Polynomial arithmetic over rings.
Sec 3.3
Sec. 4.1
Fri. 10/15 Polynomial rings over a field.
The division theorem. Divisibility
Sec. 4.1-2
Mon. 10/18 The gcd theorem
Irreducible and prime polynomials.
Sec. 4.2-3
Wed. 10/20 Sage: meet in the lab.Sage code.
The Euclidean algorithm.
Sec. 4.2-3
Fri. 10/22 Unique factorization. Sec. 4.3
Mon. 10/25 Roots and factors. Sec. 4.4
Wed. 10/27 Roots and factors. Sec. 4.5
Fri. 10/29 Congruence in F[x]. Sec. 5.1
Mon. 11/1 Congruence class arithmetic. Sec. 5.1-2
Wed. 11/3 Sage: meet in lab. Ch. 5
Fri. 11/5 Problems. Ch. 3, 4
Mon. 11/8 EXAM. Ch. 3, 4
Wed. 11/10 F[x]/m(x) for m(x) reducible and irreducible. Sec. 5.2-3
Fri. 11/12 Ideals in rings. Examples. Sec. 6.1
Mon. 11/15 Rings with only principal ideals.
New ideals from old: intersection, sum, product, radical.
Sec. 6.1
Wed. 11/17 Two rings with non-principal ideals
A ring modulo an ideal.
Sec. 6.2
Fri. 11/19 Quotient rings and homomorphisms. Sec. 6.2-3
Mon. 11/22 Some exercises: the examples from Friday, and
5.3 #9, 6.2 #25.
Sec. 6.2
Wed. 11/24 Prime and maximal ideals. Sec. 6.3
Mon. 11/29 Groups. Groups from rings. Sec. 7.1
Wed. 12/1 First properties of groups.
The order of an element.
Matrix groups
Sec. 7.1-2
Fri. 12/3 EXAM. Ch. 5, 6.
Mon 12/6 Permutation and symmetry groups. Sec. 7.1-2
Wed. 12/8 Subgroups. Sec. 7.3
Fri. 12/10 Homomorphisms and isomorphisms of groups. Sec. 7.4