Details:
Title  Computing with algebraically closed fields  Author(s)  Allan K. Steel  Type  Article in Journal  Abstract  A practical computational system is described for computing with an algebraic closure of a field. The system avoids factorization of polynomials over extension fields, but gives the illusion of a genuine field to the user. All roots of an arbitrary polynomial defined over such an algebraically closed field can be constructed and are easily distinguished within the system. The difficult case of inseparable extensions of function fields of positive characteristic is also handled properly by the system. A technique of modular evaluation into a finite field critically ensures that a unique genuine field is simulated by the system but also provides fast optimizations for some fundamental operations. Fast matrix techniques are also used for several nontrivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation.  Keywords  Algebraic closure, Algebraic number field, Algebraic function field, Field extension, Inseparability, Nonperfect field, Polynomial factorization, Root finding  ISSN  07477171 
URL 
http://www.sciencedirect.com/science/article/pii/S0747717109001497 
Language  English  Journal  Journal of Symbolic Computation  Volume  45  Number  3  Pages  342  372  Year  2010  Edition  0  Translation 
No  Refereed 
No 
