Abstract Algebra II

Prerequisites:

Abstract Algebra I

Course Description:

This course is designed for students who have completed Abstract Algebra I and are majoring in fundamental or applied mathematics. It is also open to students with a strong interest in advanced abstract structures. The objective is to deepen and extend the algebraic foundation by exploring several core advanced topics: field theory, Galois theory, commutative algebra, and an introduction to algebraic geometry. This sequence prepares students for specialized study in algebra, number theory, and geometry. Instruction will blend lectures, seminar-style discussions, and rigorous problem-solving.

Learning Objectives:

Master Field Theory. Develop a rigorous understanding of field extensions, including finite, algebraic, separable, and normal extensions, and their fundamental invariants.

Apply Galois Theory. Master the Galois correspondence and apply it to solve classical problems, including the insolvability of the quintic and constructibility questions.

Analyze Commutative Rings. Build proficiency in advanced ring theory, focusing on ideals, modules, localization, and key classes of rings like Noetherian and Artinian rings.

Connect Algebra and Geometry. Synthesize commutative algebra with the basic principles of affine algebraic geometry, understanding the dictionary between rings and geometric spaces.

Synthesize Abstract Frameworks. Integrate knowledge from all units to see the deep interconnections between group actions, ring structure, and geometric intuition.

Detailed Topics Covered:

1. Field Theory.

We begin with a detailed study of field extensions. Topics include the degree of an extension, algebraic and transcendental elements, and the construction of simple extensions. We classify extensions as finite, algebraic, separable, and normal, exploring their interrelations. Key constructions include splitting fields and algebraic closures.

2. Galois Theory.

This unit centers on the profound connection between field theory and group theory. We define the Galois group of an extension and establish the core result: the Fundamental Theorem of Galois Theory for finite extensions. This theorem sets up a precise, inclusion-reversing correspondence between intermediate fields and subgroups of the Galois group. Applications include the celebrated proof of the insolvability of the general quintic equation by radicals and results on classical straightedge-and-compass constructions.

3. Commutative Algebra.

Building deeply on ring theory, this unit explores modules over commutative rings, chain conditions (Noetherian and Artinian rings), and localization. We study the structure of ideals, including primary decomposition, and investigate how localization simplifies problems by moving to local rings.

4. Artinian Rings and Discrete Valuation Rings (DVRs).

We focus on the detailed structure of special classes of rings. For Artinian rings, we establish their characterization as finite products of local Artinian rings. This leads to the study of Discrete Valuation Rings (DVRs) as a fundamental class of one-dimensional local rings with excellent structural properties, serving as crucial examples in number theory and geometry.

5. Basic Algebraic Geometry.

The course culminates by linking algebra to geometry. We introduce affine algebraic geometry, defining affine algebraic sets and the Zariski topology. The central Nullstellensatz establishes the foundational correspondence between radical ideals and algebraic sets. This unit demonstrates how geometric objects are governed by algebraic data, synthesizing concepts from ring theory, ideal theory, and localization.