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TitleComputing with algebraically closed fields
Author(s) Allan K. Steel
TypeArticle in Journal
AbstractA 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 non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation.
KeywordsAlgebraic closure, Algebraic number field, Algebraic function field, Field extension, Inseparability, Non-perfect field, Polynomial factorization, Root finding
URL http://www.sciencedirect.com/science/article/pii/S0747717109001497
JournalJournal of Symbolic Computation
Pages342 - 372
Translation No
Refereed No