Algebraically Closed Fields (New) [HB 41]

Magma V2.7 contains an exciting new system for computing with algebraically closed fields (ACF's), 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. A similar system was suggested by D. Duval and others (the D5 system¹), 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 within the computation. These factors often arise automatically because of the structure of the algorithm which is computing over the field. The field may also be simplified and expressed as an absolute field if so desired.

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 successfully computed using an algebraically closed field.

Features:

• Automatic extension of the field by the roots of any polynomial over the field, and operations on conjugates of roots
• Basic arithmetic
• All standard algorithms for rings over generic fields work over such fields
• Minimal polynomial
• Simplification of the field
• Construction of the corresponding absolute field together with the isomorphism
• Pruning of useless variables and relations