MODERN ALGEBRA 1

Prerequisites & course information

Requisites: ( MATH 234 or 375 ), ( MATH 320 , 340 , 341 , 345 , or 375 ), and ( MATH 341 , 375 , 421 , 467 , or 521 ), 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 modern abstract algebra (e.g., groups and their actions, rings, fields, homomorphisms, standard examples of these, 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 group is abelian, whether a homomorphism is an isomorphism, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of abstract algebra. (e.g., the isomorphism theorems, the Sylow theorems, Lagrange’s theorem, 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., to classify groups of a given order, prove that alternating groups are simple, 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 areas of modern research. Audience: Graduate Learning Outcomes: 1. Recall and state the formal definitions of the mathematical objects and their properties used in modern abstract algebra (e.g., groups and their actions, rings, fields, homomorphisms, standard examples of these, 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 group is abelian, whether a homomorphism is an isomorphism, etc.). Audience: Both Grad & Undergrad 3. Recall and state the standard theorems of abstract algebra. (e.g., the isomorphism theorems, the Sylow theorems, Lagrange’s theorem, 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., to classify groups of a given order, prove that alternating groups are simple, 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 areas of modern research. Audience: Graduate