Math majors and graduate students in related areas
This is the first semester of an introduction to basic abstract algebra. It is recommended for math majors and it is essential for students preparing for graduate studies in mathematics and in some related fields.
Math 441 is a less theoretical version of Math 541
- Definition and basic properties, subgroups.
- Examples: Cyclic groups, matrix groups, unit groups, Dihedral groups, symmetric groups, etc.
- Group homomorphisms, cosets, normal subgroups, factor groups fundamental theorem of homomorphisms.
- Direct product and semi-product of groups.
- Cayley’s theorem, Lagrange theorem.
- Conjugacy classes, Sylow’s theorem, group action (if time permits).
- Definition and basic properties, subrings.
- Examples: integers, Gausian integers, Z/n, polynomial rings, matrix algebra, etc.
- Ideals, quotient rings, ring homomorphisms, fundamental theorem of homomorphisms.
- Ideals, principal ideals, integral domains, PID, maximal and prime ideals
- Irreducible polynomials in a polynomial ring, division algorithm. Unique factorization, UFD.
- Definition and basic properties, subfields.
- Examples: Q, R, C, finite fields.
- Field extensions, number fields, in particular quadratic fields and cyclotomic fields.