Math 441 - Introduction to Modern Algebra
- Prerequisites: Math 340.
- 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, 2nd ed., J. A. Beachy and W.D. Blair, Waveland Press 1996
- Course Coordinator: Arun Ram, Steven Bauman
- 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
Content coverage:
- 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).
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