Syllabus

Date Notes Topics
Jan 9 1.1 Introduction. Rings: examples and basic properties.
Jan 12 1.1–1.2 Isomorphic rings. Integral domains and fields.
Jan 16 1.2 Integral domains and fields, field of fractions
Jan 19 1.3 Ideals and quotient rings
Jan 23 1.4 Ring homomorphisms
Jan 26 1.4 First Isomorphism Theorem, Chinese Remainder Theorem.
Jan 30 2.1 Polynomial rings, division algorithm
Feb 2 2.1 Polynomial rings
Feb 6 2.2 Irreducible polynomials, factoring polynomials over the real and complex numbers, Gauss' Lemma
Feb 9 2.2 Modular Irreducibility Test, Eisenstein Criterion, greatest common divisor of polynomials
Feb 13 2.2–2.3 Euclidean algorithm, unique factorization of polynomials, quotient rings of polynomials over a field
Feb 16 2.3 Quotient rings of polynomials over a field
Feb 19–23 Reading week
Feb 27 3.1 Unique factorization domains
Mar 2 Midterm test
Mar 6 3.1 Unique factorization domains
Mar 9 3.1–3.2 Unique factorization domains, principal ideal domains
Mar 13 3.2–3.3 Principal ideal domains, euclidean domains
Mar 16 4.1–4.2 Review of vector spaces, field extensions
Mar 20 4.2 Field extensions, minimal polynomials
Mar 23 4.2 Field extensions, the Multiplication Theorem
Mar 27 4.3 Splitting fields
Apr 3 4.3 Splitting fields, algebraic closures
Apr 6 4.4 Finite fields
Apr 10 Review