Student Body:
Math majors and graduate students in related areas
Background and Goals:
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.
Alternatives:
Math 441 is a less theoretical version of Math 541
Subsequent Courses:
Math 542.
Course Content:
Group Theory
- 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).
Ring Theory
- 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.
Field Theory
- Definition and basic properties, subfields.
- Examples: Q, R, C, finite fields.
- Field extensions, number fields, in particular quadratic fields and cyclotomic fields.
credits:
3. (N-A)
semester:
FallSpring
prereqs:
MATH 341 or MATH 375 or (MATH 421 and MATH 320) or (MATH 421 and MATH 340)