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Algebra 2 (MATH 21800)

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Administrative Information

  1. Unit number and title: MATH 21800 Algebra 2
  2. Level: I/5
  3. Credit point value: 20 credit points
  4. Year: 12/13
  5. First Given in this form: 1995/6
  6. Unit Organiser: Lynne Walling
  7. Lecturer: Dr Lynne Walling
  8. Teaching block: 2
  9. Prerequisites: MATH11511 Number Theory & Group Theory and MATH 11005 Linear Algebra & Geometry

Unit aims

To develop the theory of commutative rings, and to apply it to solving problems concerning the factorisation of polynomials, algebraic numbers, ruler-and-compass constructions, and the construction of roots of polynomials.

General Description of the Unit

Algebraic structures -- such as groups, rings, and fields -- are prevasive in mathematics.  This course focuses on (commutative) rings, which are sets equipped with two (commutative) operations (called addition and multiplication), and that contain an additive identity and an additive inverse for each element of the set.  A fundamental example of a ring is Z, the set of integers; other important examples include Q,  Z modulo n, and Q[X], which is the set of polynomials in X with rational coefficients.  A fruitful way to study rings and their properties is to study "homomorphisms" between rings: a homomorphism is a map that preserves addition and multiplication (just as a linear transformation preserves vector addition and scalar multiplication).  Using homomorphisms and generalised modular arithmetic, we develop means of determining when a ring has additional nice properties, such as having multiplicitive inverses for each nonzero element of the ring.  This is a very beautiful and clean theory; in proving the theorems, the students will learn some new techniques and strengthen their proof-writing skills.

Relation to Other Units

This unit has some relationship to (but is independent of), Linear Algebra 2 and the Level 7 unit on Representation Theory, and has a stronger relationship to Algebraic Number Theory (Level 6) and Galois Theory (Level 7).   It provides the foundation for Galois Theory.

Teaching Methods

There are 3 lecture classes and 1 problems class each week. The course is based on the lectures and exercises. The basic lecture notes and exercises will be posted and distributed, and the solutions to the assigned exercises will be posted.

Solutions to most exercises not assigned will be presented in lectures, but typically these solutions will not be posted or distributed.

There are 5 in-class 10 minute quizzes (open-book, open-note). The solutions to the quizzes are discussed in the problems classes, as are common difficulties with the assigned exercises. The quizzes and homework are designed to help the students gain aptitude and confidence with the material and techniques as the term progresses.

Solutions to previous exams will not be posted or distributed.

The last 2 weeks of the course are devoted to review and revision, and in this time exercises (both assigned and not assigned) and previous exam questions will be addressed.

Besides the problems classes, there is also a weekly office hour during which students can ask questions about lectures and exercises.

Learning Objectives

After taking this unit, students should be able to state the basic definitions and results in the subject, to utilise the fundamental proof techniques, and to solve problems similar to those worked in the lectures and set as homework.

Assessment Methods

The final assessment mark for the unit will be calculated as follows:

  • 85% from a standard 2½ hour written examination in May/June consisting of FIVE questions. A candidate's best FOUR answers will be used for assessment. Calculators are NOT permitted.
  • 10% from five sets of homework questions;
  • 5% from five in-class open-book 10 minute quizzes.

Award of Credit Points

Credit points are gained by passing the unit.

Transferable Skills

The ability to understand and apply general theory, and the acquisition of facility in calculating in a variety of number-systems.

Texts

No particular text will be used, but there are many books on abstract algebra in the university library. 

These include:

"Rings, fields, and groups" by R.B.J.T. Allenby

"Contemporary abstract algebra" by Joseph A. Gallian

"A first course in abstract algebra" by John B. Fraleigh

"Abstract algebra: a first course" by Larry Joel Goldstein

 

 

 

Syllabus

Rings and subrings.

Homomorphisms, ideals, and quotient rings.

Basic homomorphism theorems.

Integral domains and fields.

Euclidean domains, principal ideal domains, and unique factorisation domains

Gauss' Lemma and consequences.

Testing polynomials for irreducibility.

Field extensions and algebraic elements.

The characteristic of a field and finite fields.

Ruler and compass constructions.