
Chapter 1
Normal Forms for Vectors and
Univariate Polynomials
In this chapter, we recall two very classical approaches to normal forms in quo-
tients, one relying on the celebrated Gaussian elimination method for solving
systems of linear equations, and the other based on the Euclidean algorithm
for computing the greatest common divisor of univariate polynomials. We
present them in a way that emphasizes some general ideas we are going to use
extensively in the rest of the book, and lay out some terminology and notation
which is used in many subsequent chapters. For both key results on normal
forms that we present in this chapter, we give two proofs,