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:

The integers, emphasizing general group and ring properties. Permutation groups, symmetry groups, polynomial rings, leading to notions of abstract groups and rings. Congruences, computations, including finite fields and applications. Emphasis on concepts and concrete examples and computations.

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 320, 340, 341 or 375 or graduate or professional standing or member of the Pre-Masters Mathematics (Visiting International) Program