This chapter gives a “look under the hood” at the algorithm that actually lets us perform computations over a polynomial ring. In order to work with polynomials, we need to be able to answer the ideal membership question. For example, there is no chance of writing down a minimal free resolution if we cannot even find a minimal set of generators for an ideal. How might we do this? If R = k[x], then the Euclidean algorithm allows us to solve the problem. What makes things work is that there is an invariant (degree), and a process which reduces the invariant. Then ideal membership can be decided by the division algorithm. When we run the univariate division algorithm, we “divide into” the initial (or lead) term. In the multivariate case we'll have to come up with some notion of initial term – for example, what is the initial term of x2y + y2x? It turns out that this means we have to produce an ordering of the monomials of R = k[x1, …, xn]. This is pretty straightforward. Unfortunately, we will find that even once we have a division algorithm in place, we still cannot solve the question of ideal membership. The missing piece is a multivariate analog of the Euclidean algorithm, which gave us a good set of generators (one!) in the univariate case. But there is a simple and beautiful solution to our difficulty; the Buchberger algorithm is a systematic way of producing a set of generators (a Gröbner basis) for an ideal or module over R so that the division algorithm works.