Gröbner bases can be used to answer fundamental questions concerning certain sets of polynomials. For example, when are two such sets equal? Given such a set and another polynomial, is this polynomial a member of that set? In this introduction to Gröbner bases, we motivate their salient features by considering the Euclidean algorithm and Gaussian row reduction of matrices from a slightly different point of view.
Flinn, Patrick, "Gröbner Bases: A Natural Extension of Gaussian Reduction and the Euclidean Algorithm (Abstract)" (2005). Kenneth C. Schraut Memorial Lectures. 6.