441 - Introduction to Modern Algebra
Prerequisites:
Math 340 or 341 or cons inst. Closed to students who have credit for Math 541.
Frequency:
Fall (I) and Spring(II)
Student Body:
Math Majors in the School of Education and some math majors in Letters and Science
Credits:
3. Students who have passed Math 541 are not permitted to take Math 441 for credit. (N-A) Recent Texts:
Abstract Algebra, An Introduction 2E, Hungerford
Course Coordinator:
Tonghai Yang Background and Goals:
This is a course in abstract algebra, but, unlike Math 541, the emphasis is on concepts and concrete examples and computations. Less proof-based than 541. Especially recommended for math majors in the school of education.
Alternatives:
Math majors in Letters and Science are recommended to take 541 instead
Subsequent Courses:
Math 541 Course Content:
- The integers, division algorithm, greatest common divisors, primes, congruences, units, well defined operations in Z mod n, field properties of Z mod p where p is prime, Euler and Fermat theorems, fundamental theorem of arithmetic.
- Cartesian products, functions, well defined, one to one, onto, composition, inverse maps, equivalence relations and partitions, inducing maps to equivalence classes.
- Groups, properties, Sym(S), subgroups, cyclic groups, LaGrange Theorem, Euler and Fermat (again), Direct Products, isomorphisms, permutation groups, even/odd permutations, motion in the plane.
- F[x] developed in parallel to E. division algorithm, gcd's, irreducibles, factor theorem, irreducibles, factor theorem, irreducibles in C[x] and R[x], polynomial congruence, extension fields, construction of roots of f(x) as in Z mod (n), finite fields.
- Rings and their properties, distributivity, construction of Q (quotient field) from Z (integral domain).