MODERN ALGEBRA 2

Prerequisites & course information

Requisites: MATH 541 , graduate/professional standing, or declared in Mathematics VISP (undergraduate or graduate) Course Designation: Core GenEd - Natural Science & Wellness Breadth - Natural Science Level - Advanced L&S Credit - L&S Liberal Arts and Science Grad 50% - Counts toward 50% graduate coursework requirement Repeatable for Credit: No Last Taught: Fall 2026 Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in advanced abstract algebra (e.g., rings, modules, fields, Galois groups, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g. whether a field extension is normal, whether a ring is a principal ideal domain, classifying abelian groups, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of advanced abstract algebra (e.g. the fundamental theorem of Galois theory, the classification of modules over principal ideal domains, results on canonical forms of matrices, etc.) and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises (e.g., determining if a polynomial is solvable over a given field, finding the normal and Jordan canonical forms of linear transformations and applying these to problems in various fields like differential equations, etc.). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in advanced abstract algebra (e.g., rings, modules, fields, Galois groups, etc.). Audience: Both Grad & Undergrad 2. Use such definitions to argue that a mathematical object does or does not have the condition of being a particular type or having a particular property (e.g. whether a field extension is normal, whether a ring is a principal ideal domain, classifying abelian groups, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of advanced abstract algebra (e.g. the fundamental theorem of Galois theory, the classification of modules over principal ideal domains, results on canonical forms of matrices, etc.) and recall the arguments for these theorems and the underlying logic of their proofs. Audience: Both Grad & Undergrad 4. Use such theorems in the context of longer arguments by examining their premises (e.g., determining if a polynomial is solvable over a given field, finding the normal and Jordan canonical forms of linear transformations and applying these to problems in various fields like differential equations, etc.). Audience: Both Grad & Undergrad 5. Prove or disprove statements related to the above definitions, properties, and theorems using techniques of mathematical argument (direct methods, indirect methods, constructing examples and counterexamples, induction, etc.). Audience: Both Grad & Undergrad 6. Convey arguments using English and appropriate mathematical terminology, notation and grammar. Audience: Both Grad & Undergrad 7. Identify applications of course content in current areas of research. Audience: Graduate