TITLE: Gröbner Basis and Division Algorithm
QUESTION [6 upvotes]: I recently read a lemma on a course in Commutative Algebra that states, 
If $G$ is a Gröbner Basis for an Ideal $I$ in $k[x_{1},...,x_{n}]$, then a polynomial $f$ belongs to $I$ if and only if $f$ on division by $G$ (we can do this using the division algorithm for polynomials) has a remainder $0$. 
My question is why would we need to go through all the trouble of calculating Gröbner basis, when we can instead say that every ideal in $k[x_{1},x_{2}...x_{n}]$ is finitely generated by $f_{1},f_{2},f_{3},...f_{s}$ (By Hilbert's Basis Theorem) and therefore divide $f$ by these $f_{i}$'s and check whether the remainder is $0$?

REPLY [3 votes]: Hilbert's basis theorem is an existence theorem. It is not constructive. We know there are basis but how to find them is a different story. Groebner basis is not about existence, it is about finding an algorithm which is efficient. Grobner basis is a generalization of Gaussian elimination where now we consider non-linear (polynomial) equations. The uniqueness of the remainder comes together with other (at list 5 more) properties (equivalences) which make the theory very solid.