Gröbner bases provide us with suitable polynomial generating sets, for ideals of $k[x,y,z,..]$ where $k$ is a field.
When applied to linear polynomials the method reduces to Gauss-Jordan elimination (for bringing a matrix to row-echelon form) and when applied in $k[x] \ $, i.e. to the ideal generated by univariate polynomials, it reduces to the usual Euclidean division in $k[x]$.
My question here is: can anyone give an account of how the method works for polynomials over $\mathbb{Z}[x]$? In other words, how can we compute the Gröbner basis of the ideal generated by two integer polynomials $f(x)$, $g(x)$?