MA20217 - Algebra 2B

Algebra 2B
Level 2

Semester 2 in 2015/16

General course information and handouts

This unit introduces students to basic abstract ring theory and provides a thorough structure theory of linear operators on a finite dimensional vector space.

Course information
Schedule of lectures
Response to Week 4 Questionnaires
Exam information (updated for the final time on 29th April, new remarks are in red).

You may also find it helpful to look at the past papers.

Class times and locations: The lectures take place on Wednesdays at 9:15 (EB1.1), Thursdays at 15:15 (EB1.1) and Fridays at 14:15 (EB1.1).

Weekly schedule and lecture notes

Week 1: Definition of ring, basic examples, subrings.
Week 2: Quotient rings and ideals (see the picture illustrating various types of ring).
Week 3: Ring homomorphisms.
Week 4: Applications of the fundamental isomorphism theorem.
Week 5: Domains, PIDs and irreducible elements.
Week 6: Unique factorisation and polynomial rings (see the picture illustrating even more types of ring).
Week 7: Algebras and field extensions.
Week 8: Number theory and minimal polynomials.
Week 9: Primary decomposition.
Week 10: Jordan normal form.

Here are the lecture notes for the whole course updated on 6th May. The only things that have changed since the final lecture are:

In Gauss' lemma and its proof, I added "in R[x]" twice to emphasise that f is primitive in R[x].
Further explanation added to the proof of the non-examinable Lemma 5.15.
Typo corrected on the penultimate line in the proof of Theorem 1.26 (what was ab is now b).
Typo corrected near the foot of p11 (what was s is now i).

Homework assignments

Please submit solutions each Thursday by 3pm to the pigeonholes near the lifts on the ground floor of 4W.

Week 1 (due in week 2): Exercises 1 with solutions (typo in solution to Ex 1.2(i) and omission in solution to Ex1.5(1) corrected in red, 29th April)
Week 2 (due in week 3): Exercises 2 with solutions.
Week 3 (due in week 4): Exercises 3 with solutions.
Week 4 (due in week 5): Exercises 4 with solutions.
Week 5 (due in week 6): Exercises 5 with solutions.
Week 6 (due in week 7): Exercises 6 with solutions.
Week 7 (due in week 8): Exercises 7 with solutions.
Week 8 (due in week 9): Exercises 8 with solutions.
Week 9 (due in week 10): Exercises 9 with solutions.
Week 10 (due in week 11): Exercises 10 with solutions.

Books that you might find useful

The lecture notes should guide you through the course. There are also a number of books that cover some or all of the content of the course, including:

A first course in abstract algebra by John B. Fraleigh. This covers much of the first two sections of the course on ring theory and factorisation.
Linear algebra by Serge Lang. This covers much of the final section of the course on the structure of linear operators.

Department of Mathematical Sciences,
University of Bath, Claverton Down,
Bath BA2 7AY.

Office: 4W 3.49
Phone: 01225 385327
Email: a.craw at bath dot ac dot uk

Algebra 2B Level 2

Algebra 2B
Level 2