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MATH504 ALGEBRA II

Course Objectives

The objective of this course is to provide a rigorous introduction to the theory of modules and field extensions, laying the foundation for advanced studies in algebra. Students will explore the structure and classification of modules over rings, including free, projective, and injective modules, as well as tensor products and modules over principal ideal domains (PIDs). The course also develops the theory of field extensions, culminating in an in-depth study of Galois theory. Topics include algebraic and normal extensions, splitting fields, algebraic closures, and the Galois group of a polynomial. Through these concepts, students will gain a deep understanding of the relationship between field theory and group theory, and how symmetries of roots of polynomials are captured by Galois groups.

Course Content

Modules; homomorphisms, exact sequences., projective and injective modules, free modules, vector spaces, tensor products, modules over a PID. Fields, field extensions, the fundamental theorem of Galois theory, splitting fields, algebraic closure and normality, the Galois group of a polynomial, finite fields.

Course Learning Outcomes

By the end of this course, students will be able to:

• Translate between algebraic problems and their module-theoretic or Galois-theoretic formulations.
• Apply abstract theory to solve computational problems involving modules, fields, and polynomials.
• Analyze and construct module homomorphisms, kernels, images, and quotient modules.
• Understand and work with exact sequences, and use them to describe structural properties of modules.
• Identify and construct free, projective, and injective modules; apply criteria for their characterization.
• Compute and interpret tensor products of modules, including applications to vector spaces.
• Analyze modules over principal ideal domains (PIDs) and apply the structure theorem for finitely generated modules over a PID.
• Define and classify field extensions, distinguishing between algebraic and transcendental extensions.
• Construct and analyze splitting fields, algebraic closures, and normal extensions.
• Understand and apply the Fundamental Theorem of Galois Theory to describe the correspondence between subfields and subgroups.
• Compute and interpret Galois groups of polynomials and understand their role in the solvability of equations.
• Explore the structure and classification of finite fields, including their construction and automorphism groups.

Program Outcomes Matrix

0: No Contribution 1: Little Contribution 2: Partial Contribution 3: Full Contribution