TITLE: how to determine whether an ideal is prime or not by an algorithm 
QUESTION [12 upvotes]: Given polynomials $f_{1},\cdots,f_{n}\in \mathbb{C}[x_{1},\cdots,x_{m}]$, do we have an algorithm to determine whether the ideal $I=(f_{1},\cdots,f_{n})$ is prime ideal or not? Of course, we assume the polynomials are irreducible.

REPLY [2 votes]: Buchberger's algorithm should do, Faugere's F4 is also one. However, this is generally for any ideal and not necessarily for irreducible. Is it something specific to irreducible polynomials that you are looking for?