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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.

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Macaulay2 has a function isPrime which tells you if an ideal is or not prime. –  Mariano Suárez-Alvarez Apr 7 '11 at 15:59
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Why do you assume that the polynomials are irreducible? –  Qiaochu Yuan Apr 7 '11 at 16:36
    
to rule out the trivial case that its factors might not be in the ideal. –  Jiang Apr 7 '11 at 17:28
    
It just doesn't seem to me that that assumption really helps. –  Qiaochu Yuan Apr 8 '11 at 4:36

4 Answers 4

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.

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Eliminate, eliminate, eliminate the eliminators of elimination theory... =) –  Harry Gindi Apr 8 '11 at 6:05

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.

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

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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?

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Irreducibility of the polynomials are not necessarily in my problem, what I need is just an algorithm to test the primality of a finite generated polynomial ideal. –  Jiang Apr 8 '11 at 3:44

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