Math 422/501

Fields and Galois Theory

Fall  2017      Exams



Midterm Exam. Friday, October 20.

Format. The exam closed book, i.e., no books, no notes, no calculators allowed. The exam is 50 minutes long, it will take place during the regular class period.

Syllabus. The exam covers all material in Chapters 1 (basic definitions and results) and 2 (splitting fields, multiple roots) of the text, except the sections on transcendental numbers and algebraically closed fields in Chapter 1. In particular, be prepared to
  • Prove that polynomials are irreducible
  • Do calculations inside a stem field
  • Decide if a number is constructible
  • Find the minimal polynomial of an algebraic element
  • Find the degree of a field extension
  • Find the degree of an element in a field extension
  • Find the splitting field of a polynomial
  • Find the number of automorphisms of a field extension
  • Decide if a polynomial is separable
The exam will consist of three parts:
  • Carefully state definitions and theorems
  • Give examples or counterexamples (no proofs)
  • Solve problems (with proofs)
Here is a sample midterm, with solutions.

Admittedly, the text is not very rich in practice problems, but a bit of rummaging on the internet will turn up plenty.

Final Exam. Monday, December 18, 15:30-18:00, location TBA.

Format. The final exam will be similar in format to the midterm exam. It will be 150 minutes long. No aids of any kind are allowed.

Syllabus. TBA

Here is a sample final.