MATH 421: Abstract Algebra II

Expands on the algebraic structure of groups from the first course in abstract algebra to introduce rings and fields. All three structures are explored via Galois theory, which shows the vital interconnectivity of the three structures, and how this can be applied to obtain deep theorems about the symmetries among roots of a polynomial. Topics include: rings, ideals, homomorphisms, polynomial rings, factorization, divisibility, vector spaces, extension fields (algebraic and transcendental), the fundamental theorem of field theory, splitting fields, classification of finite fields, constructible numbers, impossibility theorems for angle trisection and circle squaring, the fundamental theorem of Galois theory, and solvability of polynomials by radicals. Offered by Mathematics. Limited to three attempts.