MAU22102 - Fields, rings, and modules

Course information

Lecturer: Nicolas Mascot
TA: Edwina Aylward

Assessment: 15% homework assignments, 85% final exam.

Because of the current situation, all the lectures will be prerecorded and available online on this page, see below.

References

Chapters 2 and 3 of Basic algebra I by N. Jacobson cover all the contents of this module, except for the section about field extensions.
Sergey Mozgovoy's page for this module contains additional references, as well as lecture notes.

Recorded lectures

The course contents are available as prerecorded videos and as PDF notes. You can save these files by right-clicking on the links and clicking on "Save link as".

Chapter 1: Rings
Notes (Complete)
Videos:
- Rings: introduction, first examples, basic concepts (43:40)
- Fields and domains (24:24)
- Subrings (12:46)
- Ideals (35:30)
- Ring morphisms (25:19)
- Quotient rings (42:16)
- The isomorphism theorem (25:08)

Chapter 2: Arithmetic in domains
Notes (Complete, updated to include a proof of Hilbert's basis theorem)
Videos:
- The field of fractions of a domain (28:04)
- Euclidean domains (36:17)
- Principal ideal domains (18:10)
- Divisibility, associates, and irreducibles (18:12)
- Unique factorisation domains: introduction (23:22)
- Notherian rings, existence of factorisations (21:37)
- Prime ideals, maximal ideals (27:27)
- Unique factorisation domains: theorems (40:31)
- gcd and lcm in a unique factorisation domain (49:04)
- Factorisation of polynomials 1/3: over a field (13:34)
- Factorisation of polynomials 2/3: UFD[x] is a UFD (38:23)
- Factorisation of polynomials 3/3: practical results (28:15)

Chapter 3: Field extensions
Notes (Complete)
Videos:
- Field extensions, algebraicness (37:01)
- The degree of a field extension (62:04)
- Application: constructible numbers (21:21)

Chapter 4: Modules over a ring
Notes (Complete)
Videos:
- Modules vs. vector spaces (37:09)
- Module morphisms (36:37)
- Modules over a PID: theorems (34:14)
- Modules over a PID: applications (44:13)

Homework

You must submit your homework assignments before the deadline, by emailing them to the TA.

Typesetting your work in LaTeX will increase your mark by 20% (but it will still be capped at 100/100).

The solutions will be made available immediately after the deadline, so late submissions will not be accepted.

Exercise sheet	Due date	Solutions	LaTex source
Sheet 1	Friday February 19th, 4PM	Solutions 1	Source 1
Sheet 2	Friday March 5th, 4PM	Solutions 2	Source 2
Sheet 3	Monday March 22nd, 4PM	Solutions 3	Source 3
Sheet 4	Tuesday April 6th, 4PM	Solutions 4	Source 4
Sheet 5	Monday April 19th, 4PM	Solutions 5	Source 5

Final exam

The final exam will probably take place in the form of a take-home exam. A mock exam paper will be provided the real exam, so you can practice and prepare yourself.

Back to Nicolas Mascot's home page