Math 81/111: Abstract Algebra (Rings and Fields)

Winter 2015


Course Info:



[PDF] Syllabus

This course provides a foundation in core areas in the theory of rings and fields. Specifically, it provides an introduction to commutative ring theory with a particular emphasis on polynomial rings and their applications to unique factorization and to finite and algebraic extensions of fields. The study of fields continues with an introduction to Galois Theory, including the fundamental theorem of Galois Theory and numerous applications.



The homework assignments will be assigned on a weekly basis and will be posted below. Homework is due in one week; no late homework will be accepted.

Cooperation on homework is permitted (and encouraged), but if you work together, do not take any paper away with you--in other words, you can share your thoughts (say on a blackboard), but you have to walk away with only your understanding. In particular, you must write the solution up on your own. Please acknowledge any cooperative work at the end of each assignment.

Plagiarism, collusion, or other violations of the Academic Honor Principle, after consultation, will be referred to the The Committee on Standards.

[PDF] Homework Submission Guidelines

15 Jan(M)Introduction
27 Jan(W)1: Rings, fieldsHW 1: PDF and TeX (updated 7 Jan); Solutions
38 Jan(R)Review of Euclidean domains, PIDs, UFDs
49 Jan(F)1: Polynomial rings, factoring, extension fields
512 Jan(M)1: Algebraic and transcendental elements
614 Jan(W)1: Constructions with straight-edge and compassHW 2: PDF and TeX; Solutions
715 Jan(R)1: Algebraically closed fields
816 Jan(F)2: Splitting fields
-19 Jan(M)No class, Martin Luther King day
921 Jan(W)2: Multiple rootsHW 3: PDF and TeX; Solutions
1022 Jan(R)Quadratic fields, examples
1123 Jan(F)2: Separability, 3: Automorphisms of fields
1226 Jan(M)3: Separable, normal, Galois extensions
1328 Jan(W)3: Fundamental theorem of Galois theory, examplesHW 4: PDF and TeX (updated 29 Jan); Solutions
1430 Jan(F)3: Fundamental theorem of Galois theory, proofs
152 Feb(M)3: Fundamental theorem of Galois theory, consequences
-4 Feb(W)No class, JV in RochesterHW 5: PDF and TeX (updated 10 Feb); Solutions
165 Feb(R)3: Examples
-6 Feb(F)No class, Winter carnival
179 Feb(M)3: Constructible numbers, Galois group of a polynomial;
4: A_n, transitive groups
1811 Feb(W)4: Cubic and quartic polynomials, Finite fieldsHW 6: PDF and TeX; Sage code; Solutions
1912 Feb(R)4: Computing Galois groups over QQ
2013 Feb(F)5: Primitive element theorem
2116 Feb(M)5: Hilbert's Theorem 90
2218 Feb(W)5: Kummer theoryHW 7: PDF and TeX; Solutions (simplified solution to 7.3, 11 Mar)
2319 Feb(R)6: Algebraic closures
2420 Feb(F)5: Solvable extensions
2523 Feb(M)7: Topological groups, Krull topology
2625 Feb(W)7: Fundamental theorem of infinite Galois theoryHW 8: PDF and TeX; Solutions
2827 Feb(F)7: Fundamental theorem of infinite Galois theory, proofs
-2 Mar(M)No class, JV in Bonn
-4 Mar(W)No class, JV in Bonn
296 Mar(F)Guest lecture by Peter Doyle
309 Mar(M)Review and wrap-up
*13 Mar(F)Final Exam, 8:00 - 11:00 a.m.; Solutions