Question

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.

Answer 1

Let $R$ be a Noetherian ring and let $I$ be an ideal in $R[x]$. Then the following facts hold:

• $I$ is prime in $R[x]$ $\Longleftrightarrow$ $I\cap R$ is prime in $R$ and $\overline{I}$ is prime in $R/(R\cap I)$.

• If $R$ is an integral domain and $I \cap R=0$, then $I$ is prime in $R[x]$
  $\Longleftrightarrow$ $I K[x]$ is prime in $K[x]$ and $I=IK[x]
  \cap K[x]$. Here $K$ denotes the fraction field of $R$.

Using the above to successively eliminate variables, this shows that one can reduce the problem of checking primiality to the one-variable case, where many efficient methods are known. I think this is also how the Grobner basis works, since these can algorithmically compute the elimination ideals above.

Answer 2

There is such a test. Some explanation can be found in: "An introduction to Gröbner bases, By William Wells Adams, Philippe Loustaunau", or the original article (http://portal.acm.org/citation.cfm?id=65034) the above text is based on. See also the singular manual.

Answer 3

Did you look into "A Singular Introduction to Commutative Algebra"?

Answer 4

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?