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Magma contains a system [Ste02], [Ste10] for computing with
algebraically closed fields, which have the property that they always
contain all the roots of any polynomial defined over them.
It is of course 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.
Such a system was already proposed before (the D5 system [DDD85]),
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.
Magma's system has no such problem and one can compute with the field
just like any other field in Magma; all standard algorithms which work
over generic fields or which use factorization work automatically without
having to be adapted to handle the many conjugates of a root.
Especially significant is also the fact that all the Gröbner basis
algorithms work well over such fields. One can compute the variety
of any zero-dimensional multivariate polynomial ideal over the
algebraic closure of its base field. Puiseux expansions of polynomials
are also successfully computed using an algebraically closed
field.
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