Simplifying an algebraic integer expression

I have an expression where the variables are algebraic integers: $p4 = \frac{p12 - p41 \cdot p21}{p22}$ p12 is degree 48 and p22 is most likely degree 48 too. p41 is degree 32 and p21 is degree 24. I am trying to avoid a geometric increase in degrees, p41*p21 is 32*24 = 768 and p12 - p41*p21 would be 48*768 = 36864 and then dividing this by p22 would give 48*36864 = degree 1769472 would is well beyond the capability of my computer (using GP-Pari)

How can this be simpified? I strongly suspect that p42 is degree 192, but have no way to determine this unless I grind it out using the algdep command of GP-Pari which depends upon the accuracy of the digits (which will have to be around 20K) This would take several days to run.

Perhaps you have more info that could reduce the degree. For example, the fields generated individually by $p_{41}$ and $p_{21}$ could intersect in a field of degree 16. and similar things.Also possibly a a convenient automorphism (a Galois group element) might send your number $p_4$ to a number more amenable for computations. – P Vanchinathan May 14 '13 at 6:19
@Randall: Isn't your question essentially the same one as your question mathoverflow.net/questions/24513/… from 3 years ago? How are your numbers $pxy$ given, by minimal polynomials or complex floats up to some precision? – Peter Mueller May 14 '13 at 16:05