MATH 302, Abstract Algebra II

Overview We will focus on the theory of groups, abstract structures that are more general than rings, domains, and fields, which were studied in Abstract Algebra I. Because groups have fewer axioms than rings, they are more diverse and have applications to a broader range of topics. The first half of the semester will be devoted to group theory itself; afterwards we will briefly review the theory of polynomial rings, vector spaces, and field extensions so that we can study Galois theory which ties many of these ideas together. Finally we will study applications of abstract algebra to coding theory and/or classical questions of constructibility of geometric objects and solvablility of polynomial equations by radicals.

Course Documents

• Syllabus and Tentative Semester Plan
• Day-by-day plans, with assignments: Unit 1, Unit 2, Unit 3, Unit4
• Hints and Modifications to Homework Problems

Reading Questions

• Section 1.2, Sets and Equivalence Relations
• Section 3.1, Integer Equivalence Classes and Symmetries
• Section 3.2, Groups: Definitions and Examples
• Section 3.3, Subgroups
• Section 4.1, Cyclic Subgroups-I
• Section 4.1, Cyclic Subgroups-II
• Section 5.1, Permutation Groups: Definitions and Notation
• Section 5.2, Dihedral Groups
• Section 6.1, Cosets
• Sections 6.2-3, Lagrange's, Fermat's, and Euler's Theorems
• Section 9.1, Isomorphisms: Definitions and Examples
• Section 9.2, Direct Products
• Section 10.1, Normal Subgroups and Factor Groups
• Section 11.1, Group Homomorphisms
• Section 11.2-I, The First Isomorhism Theorem
• Section 11.2-II, The Second and Third Isomorhism Theorems
• Section 13.1, Finite Abelian Groups
• Section 14.1, Group Actions
• Section 14.2, The Class Equation
• Section 15.1-I, The First Sylow Theorem
• Section 15.1-II, The Second and Third Sylow Theorems
• Section 15.2, Examples and Applications
• Sections 16.1-16.2, Rings, Integral Domains, and Fields
• Section 16.3, Ring Homomorphisms, Ideals
• Section 16.4, Maximal and Prime Ideals
• Sections 17.1-17.2, Polynomial Rings, The Division Algorithm
• Section 17.3, Irreducible Polynomials
• Section 21.1, Extension Fields
• Section 21.2, Splitting Fields
• Section 21.3, Geometric Constructions
• Section 23.1, Field Automorphisms
• Section 23.2, The Fundamental Theorem of Galois Theory
• Section 13.2, Solvable Groups
• Section 23.3, Applications of Galois Theory

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