Linear algebra—solving linear systems

Systems of linear equations over a field K can be solved by various methods, e.g., by Gaussian elimination or Cramer’s rule. But if we start with a system over the integers, we will immediately introduce rational numbers, whose arithmetic operations are clearly more costly than the corresponding operations on integers. So for the same reason as in the computation of gcds of polynomials or factorization of polynomials, we are interested in a method for solving systems of linear equations which avoids computation with rational numbers as much as possible. Such a method for fraction free Gaussian elimination is Bareiss’s algorithm, as described in Bareiss (1968).