Department of Mathematics Syllabus
This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.
MAT 250A: Algebra
Approved: 2015-11-01 (revised 2025-05-30, Fuchs and Vazirani)
Units/Lecture:
Every year; 4 units; lecture/term paper or discussion
Suggested Textbook: (actual textbook varies by instructor; check your
instructor)
David Dummit, Richard Foote, Abstract Algebra
Prerequisites:
Graduate standing or permission through Student Services..
Suggested Schedule:
| Lectures | Topics/Comments |
|---|---|
| 1-2 | Brief review of quotient groups. Isomorphism theorems. |
| 2 | Composition series: existence for finite groups, examples, Jordan Holder. |
| 2 | Abelian and cyclic series, solvable groups, derived series, central series. |
| 1-2 | Group actions: main theorems and, if desired, G-set homomorphisms. |
| 1 | Automorphism groups, inner automorphisms, examples. |
| 1-2 | Internal and external semidirect products. |
| 2-3 | Sylow theorems and applications to classifying certain groups of finite order. |
| 1 | Finitely generated abelian groups (see note) |
| 1 | Algebraic field extensions, degree, compositum of fields. |
| 1 | Splitting fields |
| 1-2 | Algebraic closure and algebraically closed fields |
| 2 | Separability, finite fields. Algebraic closure of a finite field. |
| 1-2 | Automorphism groups of field extensions, L-valued characters and Dedekind's Lemma. |
| 2 | Galois extensions and Galois groups, Artin's theorem. |
| 3-4 | Galois correspondence, solvability of polynomials, cubic formula. |
| 1 | Geometric constructions (see note) |
Additional Notes:
All of the topics above are easily found in Dummit and Foote, and students may benefit from the ampleness of examples in this book. However, especially for the instructor, it may be better to use smaller and more topic-specific books and notes to get across the big picture for the topics covered.
A good source for group theory topics is Groups and Representations by J.L. Alperin and Rowen Bell, chapters 1,3, and 4. Most of the field theory is beautifully covered in the classic book Galois Theory by Emil Artin, although one has to supplement this with another source (such as Dummit and Foote or J.S. Milne's online Field and Galois Theory notes) for topics like algebraic closure and composite fields.
Classification of finitely generated abelian groups is important to state, though the proof can be deferred to 250B, where the more general case of finitely generated modules over a PID is handled.
For a self contained, relatively short proof similar to Kronecker's original proof, a good source is J.S. Milne's Group Theory notes. There is likely not time for both the cubic formula and geometric (ruler-compass) constructions. The instructor can choose one of these topics in that case.
23-31 Lectures