Mathematica 9 covers polynomial algebra over finite fields. I have zero experience with this part of Mathematica, but I would guess that it would at least provide a reasonable start. They have instructive examples that should allow you to see whether their scope matches your needs.
The best are Cricket graph and mathematica. These deal with the polynomials in most of the fields.
Hi Patrick
I would strongly recommend Singular: http://www.singular.uni-kl.de/
i hAVE USED IT FREQUENTLY AND ITS FREE AND WELL ESTABLISHED.
Of course you might want to try also, Macaulay2 or CoCoa (Also free)
Then there are of course commercial programs like MAPLE OR MATHEMATICA.
If you have problems with singular I might provide you a code
where a system is solved over some small characteristic.
regards
Samuli
SAGE and Singular are open source softwares which have many toolboxes and utilities. The polybori library of SAGE can compute a Grobner basis of a reasonably small degree systems over fields of small char. For large finite fields and large degrees Grobner basis method fails due to memory req. In applications such as cryptology equations are often in extension fields of char 2. Hence in such cases you may also use the Boolean approaches. In my experience Boolean approach is more scalable than Grobner basis approach.
I would recommend Singular or Magma.
Whether you need to understand GBs depends a lot of the size and difficulty of your system of polynomials. You may find that you can get a solution just using the default settings. If you are on the edge of what's feasible, knowing about GBs could help you to choose settings that will speed up the computation significantly (for example there are faster methods for particular kinds of system, such as characterist 2 or degree 2). But the known algorithms are not polynomial time, so you soon get to systems that can't be solved with existing methods.
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