Faculty of Science Course Syllabus
Department of Mathematics & Statistics
MATH 4055/5055: Advanced Algebra II (Winter 2025)
Instructor: Theo Johnson-Freyd, theojf@dal.ca.
Lectures: Tuesdays and Thursdays, 10:05-11:25 Atlantic Time, Chase 227.
Office Hours: Wednesdays, 14:35-16:55 Atlantic Time, Chase 214.
Course website: http://categorified.net/25Winter5055/.
This course is governed by the academic rules and regulations set forth in the University Calendar and by the Senate.
Course description
A field is a commutative ring with no proper ideals. It follows from the definition that any homomorphism F → E of fields is an injection aka an extension. This course will study the theory of field extensions, equivalently the category of fields.
Quite remarkably, we will associate to each field F a group Gal(F) in such a way that the category of field extensions of F is very well approximated by the category of sets with a transitive Gal(F)-action. (Here "very well approximated" means "up to adding appropriate finiteness conditions".) This allows questions of commutative algebra and module theory to be translated into questions of group theory.
There are various qualitative "sizes" of extensions F → E. The ones controlled by Gal(F) are the "smallest", called the separable extensions. There are also finite but purely inseparable extensions, and we will analyze these — they are an interesting positive-characteristic phenomenon. Finally, there are purely transcendental extensions, which have the flavour of modules or vector spaces, including a theory of bases.
Course materials and delivery
The main textbook for the class is:
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Abstract Algebra by D. Dummit and R. Foote, chapters 13 and 14.
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Algebra by T.W. Hungerford, chapters V and VI.
Algebra by S. Lang, chapters V, VI, and VII.
Galois Theory by E. Artin, chapters 2 and 3.
Homework, notes, and so on will be available on the Course Content page.
Lectures will be given in person. (The occasional lecture may be delivered over Zoom.) This is a small class. Although attendance is not mandatory, your presence or absence will certainly be noticed by all participants.
Course Assessment
Homework assignments will be distributed roughly every two weeks. I encourage you to work together to complete your homework assignments. Studies have shown that social ties are a main predictor of success in STEM classes. Although you are encouraged to work together, the homework you submit must be written by you individually. There are many online resources offering solutions to homework at all levels. If you choose to use such resources, please be cautious: they often provide too detailed an answer, and students can trick themselves into thinking that they understand more than they do by copying those answers.
There will be a two-hour final exam at the end of the semester. Graduate students have the option to instead take the three-hour Algebra Comprehensive Exam Part II, which will consist of the final exam together with one or two additional questions on material from Math 3031. There will not be a midterm.
Each graduate student will also give a presentation (roughly 15 minutes) on a topic related to the course, to be selected in consultation with the instructor. Undergraduate students are invited to give a presentation for extra credit. Some topics people have spoken on in the past: The inverse Galois problem (engineering specific Galois groups); Galois connections in orders logic (on the abstract theory of the Galois correspondence); and Galois theory for graphs (combining Galois theory, combinatorics, and algebraic topology). Presentations will be given throughout the course.
Component weighting
Math 4055:
Homework: 60%.
Final exam: 40%.
Math 5055:
Homework: 50%.
Presentation: 20%.
Final exam: 30%.
Conversion of numerical grades to Final Letter Grades follows the Dalhousie Common Grade Scale. Note that for graduate students, any grade below B- is recorded as an F.