Department of Mathematics

Van Vleck Hall, 480 Lincoln Drive, Madison, WI

Math 441: Introduction to Modern Algebra

 
Student Body: 

Especially recommended for option 2 majors intending to become high school math teachers. See the Secondary Education variant of the option 2 sample packages.

Suitable to anyone who wants to know a little modern algebra after Math 341 but does not want to get really abstract.

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.

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, congruence, 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, maps, equivalence relations and partitions, inducing maps to equivalence classes.
  • Groups, properties, subgroups, cyclic groups, LaGrange Theorem, Euler and Fermat , Direct Products, isomorphism, permutation groups, even/odd permutations, motion in the plane.
  • F[x] developed in parallel to integers. division algorithm, gcd's, irreducibles, factor theorem, polynomial congruence, extension fields, construction of roots of f(x) as in Z mod (n), finite fields.
  • Rings and their properties, Gaussian integers, matrix rings, polynomial rings.
  • Fields and their properties, finite fields, examples of number fields.

Keep in mind the following connection with high school mathematics.

  • The isomorphism between the additive group of the real numbers and the multiplicative group of the positive real numbers given by the exponential and logarithmic functions.
  • Compare the arithmetic in Z and Z/nZ to understand the importance of the lack of zero divisors when solving polynomial equations by factoring.
  • Connect the algebra of polynomial rings and the base 10 arithmetic of integers.
  • Distinguish polynomials from polynomial functions (and show that these rings are isomorphic over an infinite field).
  • Discuss the existence of algorithms for solving cubics and quartics, but not for polynomials of higher degree.
  • The rational root theorem for integer polynomials.
  • A brief discussion of compass and straightedge constructability and the three classical "hard" problems: trisecting an angle, squaring a circle, doubling a cube.
  • RSA algorithm.
credits: 
3. (N-A)
semester: 
FallSpring
prereqs: 
Math 340 or 341 or consent of instructor. Closed to students who have credit for Math 541.

UW-Madison Department of Mathematics
Van Vleck Hall
480 Lincoln Drive
Madison, Wi  53706

(608) 263-3054

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