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, 15:3018:00, 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 16 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.
