This article proposes a new method for finding certain solutions ol systems of algebraic equations, a method successfully applied to huge systems where general methods usually fail. We use p-adic approximation, and combine a trivial but effective method for solving systems modulo p that takes advantage of a small p with a new lifting scheme able to cope with bad reduction, this being a frequent phenomenon with small primes. The method works particularly good for solutions of small (or at least bounded) algebraic degree. Such solutions are interesting in Constructive Galois Theory where they yield polynomials with a prescribed Galois Group.