ABSTRACT
In this chapter, we recall two very classical approaches to normal forms in quotients, 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, a theoretical one which proves existence of something without a specific computational recipe, and a constructive one which gives an algorithm one can use to achieve the goal. Throughout the book, we aim to keep a certain balance between the two approaches: theoretical applications are always our goal, but actual computations sometimes end up being the key to them, and as such cannot be dismissed. Knowing that something exists is always useful, but knowing how to construct it may be even more useful.
