is reducible the necessary condition for finding an ideal?

i use combination to choose a number of terms added all in a list of terms in invariant ring
to see whether it is able to factor, if it can be factored then print it

is reducible or spit field the necessary condition before finding an ideal?

if there is N * N permutation matrix from finite group , is choosing N+1 that is the correct number of terms to be factorized?