Math majors in Letters and Science are recommended to take 541 instead
- 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.