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Faculty of Science

MATH 4055/5055: Advanced Algebra II

Theo Johnson-Freyd, theojf@dal.ca
Lectures: TuTh 11:35-12:55.
Office hours: We 14:35-16:55.

Faculty of Science Course Syllabus
Department of Mathematics & Statistics

MATH 4055/5055: Advanced Algebra II (Winter 2024)

Instructor: Theo Johnson-Freyd, theojf@dal.ca.
Lectures: Tuesdays and Thursdays, 11:35-12:55 Atlantic Time, Chase 227.
Office Hours: Wednesdays, 14:35-16:55 Atlantic Time, Chase 214.
Course website: http://categorified.net/24Winter5055/.

This course is governed by the academic rules and regulations set forth in the University Calendar and by the Senate.

About this course

A field is a commutative ring with no proper ideals. It follows from the definition that any homomorphism FE 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 FE. 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

The main textbooks for the class are:

It is recommended but not required that students own copies of both books.

Homework, notes, and so on will be available on the Course Content page.

Course Delivery

Lectures will be given in person. (The occasional lecture may be delivered over Zoom.) This is a very small class. Although attendance is not mandatory, your presence or absence will certainly be noticed by all participants. Lectures will not be recorded.

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 student will also give a presentation (roughly 15 minutes) on a topic related to the course, to be selected in consultation with the instructor. Some topics people have spoken on in the past: Burnside's Lemma (a link between algebra and combinatorics), What do free groups and group rings have in common? (the use of category theory in algebra), and The non-existence of a construction for trisecting an angle (a link between algebra and geometry). Presentations will be given throughout the course.

Component weighting

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.