Algebraically closed fields (ACF's) have the property that they always contain all the roots of any polynomial defined over them.
It is not possible to construct explicitly the closure of a
field, but the system works by automatically constructing larger and
larger algebraic extensions of an original base field as needed during
a computation, thus giving the illusion of computing in the algebraic
closure of the base field.
A similar system was suggested by D. Duval and others (the D5 system [7]), but this has difficulty with the parallelism which occurs when one must compute with several conjugates of a root of a reducible polynomial, leading to situations where a certain expression evaluated at a root is invertible but evaluated at a conjugate of that root is not invertible. The scheme developed for Magma by Allan Steel avoids these problems. Consequently, ACF's behave in the same way as as any other field implemented in Magma; all standard algorithms implemented for generic fields and which use factorization work without change (for example, the Jordan form of a matrix). The new system avoids factorization over algebraic number fields when possible, and automatically splits the defining polynomials of a field when factors are found. The field may also be simplified and expressed as an absolute field. Especially significant is also the fact that all the Gröbner basis algorithms work well over ACF's. One can now compute the variety of any zero-dimensional multivariate polynomial ideal over the algebraic closure of its base field. Puiseux expansions of polynomials are now also computed using an algebraically closed field.