MAS 5311
ABSTRACT ALGEBRA I
Fall 2009

Instructor: Amod Agashe Email: agashe@math.fsu.edu
Office: 216 LOV Phone: 644-8704
Web page: http://www.math.fsu.edu/~agashe/grv3.html
Office Hours: To be announced.

Recommended text. D. Dummit, R. Foote, Abstract Algebra, Wiley.
Course Content. This course is the third in a three semester sequence (the first two being Groups, Rings, and Vector spaces I and II), whose goal is to cover the topics on the algebra qualifier (listed below), and some additional topics if time permits. We will pick up from where we left in GRV II, roughly in the middle of the topic "Fields", which is in the "More advanced topics" section below. We will most likely have time for some additional topics, about which I will discuss in class.
    Basic topics:
    • Groups:
      • Basic definitions; subgroups; cosets; normal subgroups;
      • quotient groups; homomorphisms; fundamental theorems on group homomorphisms.
    • Rings:
      • Basic definitions; ideals; ring homomorphisms; fundamental theorems on ring homomorphisms;
      • Prime ideals, maximal ideals; quotient rings;
      • fields.
    • Modules and Vector spaces:
      • Basic definitions; submodules; quotient modules; homomorphisms; fundamental theorems on module homomorphisms;
      • direct sums; free modules;
      • Vector spaces; subspaces; quotient spaces; linear transformations;
      • matrices; rank and nullity, dual vector spaces, determinants and non-singularity;
      • eigenvectors and eigenvalues; characteristic polynomials;

    More advanced topics:
    • Groups:
      • permutation groups; Cayley's theorem;
      • actions of a group on a set.
      • Sylow's theorems;
      • Free groups
      • Fundamental theorem of finitely generated abelian groups (statement only)
    • Rings:
      • Euclidean domains; principal ideal domains, unique factorization domains;
      • Polynomial rings, Gauss' lemma;
    • Fields:
      • Finite, algebraic, simple, normal, separable and inseparable extensions; splitting fields; finite fields;
      • Galois correspondence; fundamental theorem of Galois theory; solvability of equations by radicals
    • Modules and vector spaces:
      • classification of finitely generated modules over PIDs;
      • minimal polynomials; Cayley-Hamilton theorem;
      • Rational and Jordan canonical forms;
Grading. To be announced in class.
Exam Policy. No makeup exams will be given. An absence from an exam may be excused if the student presents sufficient evidence of extenuating circumstances and gets permission BEFORE the exam (unless it is an emergency). Absences due to family social events will not be excused. If a test absence is excused, the final exam grade will be used in its place.
Honor Code. The Academic Honor System at The Florida State University is based on the premise that each student has the responsibility 1) to uphold the highest standards of academic integrity in the student's own work, 2) to refuse to tolerate violations of academic integrity in the University community, and 3) to foster a high sense of integrity and social responsibility on the part of the University community. A copy of the University Academic Honor Code can be found in the current Student Handbook and you are bound by it in all your academic work.
American Disabilities Act. Students with disabilities needing academic accommodations should register with and provide documentation to the Student Disability Resource Center (SDRC), and bring a letter from the SDRC to the instructor indicating their needs.This should be done within the first week of class.