Course Content
Approximate agenda
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Field extensions. Degree of an extension. Finite-degree extensions are algebraic. Compass and straightedge constructions.
Splitting fields. Algebraic closure. Normality. Galois extensions.
The fundamental theorem of Galois theory.
Galois groups of polynomials. Symmetric functions.
Cyclotomic extensions. Abelian extensions. Kummer theory.
Radical extensions. Solvability.
Finite fields.
Galois groups over Q.
Inseparable extensions.
Transcendental extensions.
Homework assignments
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Assignment 1, due 23 January 2025.
Assignment 2, due 6 February 2025.
Assignment 3, due 25 February 2025.
Assignment 4, due 18 March 2025.
Exams
The final exam / Algebra II comprehensive exam will consist of nine questions. Students taking just the final exam are expected to choose six questions of the nine questions, and will have two hours for the exam. Students taking the comprehensive exam are expected to choose eight of the nine questions, and will have three hours for the exam. Questions will be similar to, and possibly identical to, questions on the following samples: