Modern Algebra II

An introduction to the theory of rings and fields, including the following topics: Rings; subrings; units in rings; fields; integral domains; homomorphisms and isomorphisms of rings; the field of fractions of an integral domain; polynomial rings; the division algorithm for polynomials; ideals and congruence; quotient rings; prime and maximal ideals; kernels of ring homomorphisms; the characteristic of a ring; the universal property of the quotient ring; the First Isomorphism Theorem for rings; irreducible polynomials and roots of polynomials; unique factorization of polynomials; algebraically closed fields; vector spaces over a field; field extensions; algebraic and transcendental elements; finite extensions and algebraic extensions; the minimal polynomial of an algebraic element; simple extensions; algebraic closure. Additional topics may include: Straightedge and compass constructions; the constructible numbers; principal ideal domains, unique factorization domains, and Euclidean domains; splitting fields; separable extensions; Galois groups, Galois extensions, and the Fundamental Theorem of Galois Theory; modules over rings. Credit cannot be held for both this course and 62:331.