matthias beck

MATH 850: Algebra Spring 2023

Course objectives. Algebra studies the structure of sets with operations, such as integers with addition and multiplication, or vector spaces with linear maps. The abstract point of view, based on an axiomatic approach, reveals many deep ideas behind seemingly innocent structures--such as the arithmetic of counting numbers--and serves as an elegant organizing tool for the vast universe of modern algebra. Generations of brilliant minds have crystallized these ideas in the concepts of groups, rings, fields, modules, and their quotient structures and homomorphisms--the topics of MATH 335 & 435/735. Building on this foundation, our main goal in Math 850 is to study three areas of modern algebra whose common theme is polynomials: Gröbner bases, Galois theory, and monomial ideals & simplicial complexes.

Syllabus. Polynomial rings, irreducibility criteria, Gröbner bases & Buchberger's algorithm, field extensions, splitting fields, Galois groups, fundamental theorem of Galois theory, applications of Galois extensions, squarefree monomial ideals, Hilbert series, simplicial complexes and homology, free resolutions and Betti numbers.

Textbook. David S. Dummit & Richard M. Foote, Abstract Algebra (3rd edition), Wiley 2004. [errata]

Homework & grading system. I will assign homework problems as we go through the material. You may (and should) work together with your class mates. We can discuss the homework problems at any time during class, and you can hand in any of your solutions for feedback. We will have a homework quiz every Friday at the beginning of class, in which you will be asked one definition and one problem given in the previous week. There will be no other exams.

• Gröbner basis homework
• Combinatorial commutative algebra homework

I want to ensure that each of you accomplishes the goals of this course as comfortably and successfully as possible. At any time you feel overwhelmed or lost, please come and talk with me.

Sage. You will be expected to use the open math software sage in some of your homework assignments. Here is a good introduction to sage.

The math. The way to learn math is through doing math. It is vital and expected that you attend every lecture. You will get a good feel for the math from there, but it is even more crucial that you do the homework. Working in groups is not only allowed but strongly recommended. Our class is based on Federico Ardila's Axioms:

1. Mathematical potential is distributed equally among different groups, irrespective of geographic, demographic, and economic boundaries.
2. Everyone can have joyful, meaningful, and empowering mathematical experiences.
3. Mathematics is a powerful, malleable tool that can be shaped and used differently by various communities to serve their needs.
4. Every student deserves to be treated with dignity and respect.

Fine print.
SFSU academic calender
BS rule
Academic Integrity and Plagiarism
Tutoring
CR/NCR grading
Incomplete grades
Late and retroactive withdrawals
Students with disabilities
Religious holidays

This syllabus is subject to change. All assignments, as well as other announcements on tests, policies, etc., are given in class. If you miss a class, it is your responsibility to find out what's going on. I will try to keep this course web page as updated as possible, however, the most recent information will always be given in class. Always ask lots of questions in class; my courses are interactive. You are always encouraged to see me in my office.