Field Theory: Let a = 3 + 2^1/3 + 2 * 2^2/3. We show that 1) Q[a] = Q[2^1/3], 2) the minimal polynomial of a over Q is x^3 - 9x^2 + 15x - 25, and 3) a^-1 expressed as a polynomial in a over the rationals. The main technique from linear algebra is the minimal polynomial of a linear transformation.
Includes course on Group Theory (problems and solutions at website) and Ring Theory, and Field Theory. For Prerequisites on proofs and sets, see the Math Major Basics course.