TITLE: Solving polynomial equations when you know in which number field the solutions live
QUESTION [16 upvotes]: Suppose I have a bunch of polynomial equations with coefficients in a number field, and suppose further that I'm guaranteed a priori that they have a solution in that number field.  Can I leverage that knowledge into a technique for solving the equations more easily?  The cases we care about are massively overdetermined systems of linear and quadratic equations.

REPLY [2 votes]: One minor comment: if you can ever reduce to solving equations in a single variable, and looking for rational roots, then you can use the rational root theorem. You can reduce to the case where your number field is the rationals easily enough. (For example, if you started out looking for roots in Q(i), write everything as a+bi.) But I don't know that there is any good way to reduce to equations in a single variable.