  Large linear systems play an important role in high-energy theory, appearing
in amplitude bootstraps and during integral reduction. This paper introduces
FiniteFieldSolve, a general-purpose toolkit for exactly solving large linear
systems over the rationals. The solver interfaces directly with Mathematica, is
straightforward to install, and seamlessly replaces Mathematica's native
solvers. In testing, FiniteFieldSolve is approximately two orders of magnitude
faster than Mathematica and uses an order of magnitude less memory. The package
also compares favorably against other public solvers in FiniteFieldSolve's
intended use cases. As the name of the package suggests, solutions are obtained
via well-known finite field methods. These methods suffer from introducing an
inordinate number of modulo (or integer division) operations with respect to
different primes. By automatically recompiling itself for each prime,
FiniteFieldSolve converts the division operations into much faster combinations
of instructions, dramatically improving performance. The technique of compiling
the prime can be applied to any finite field solver, where the time savings
will be solver dependent. The operation of the package is illustrated through a
detailed example of an amplitude bootstrap.
