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, Math 102.

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. The exam covers all material from Chapters 1-6 of the textbook, with the following exceptions:
  • Transcendental numbers
  • Quartic polynomials (substitute what we learned about biquadratic polynomials)
  • Computing Galois groups over Q (except for Proposition 4.26, which you do need to be familiar with)
  • Fundamental Theorem of Algebra
  • Normal basis theorem
  • Hilbert 90
  • Symmetric polynomials
  • General polynomials
  • Norms and traces
  • Etale algebras
  • Separable closures
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 Galois group of a polynomial
  • Especially of a polynomial of degree less than 5
  • Find the Galois group of a field extensions
  • Exhibit the lattice of intermediate fields of a field extension using the fundamental theorem of Galois theory
  • Find primitive elements of field extensions
As the midterm exam, the final will consist of three parts:
  • Carefully state definitions and theorems
  • Give examples or counterexamples (no proofs)
  • Solve problems (with proofs)
The textbook has review excercises in Appendix A, with solution hints in Appendix C.

Here is a sample final.