prime ideal factorization in Qbar or C,

[Soroosh]

Hi,

does anybody have a suggestion on how one goes about doing a primary
decomposition of an ideal over C (or Qbar)? As a concrete example, the
ideal generated by
p_1 q_1 - q_4 p_4 = 0,
p_2 q_2 - q_1 p_1 = 0,
p_3 q_3 - q_2 p_2 = 0.

is prime over Q[p_i,q_i]. How can I prove if it is (or is not) prime
(or primary) over Qbar?

Thanks in advance,
Soroosh