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In this paper we analyze the computational costs of various operations and algorithms in algebraic number fields using exact arithmetic. Let K be an algebraic number field. In the first half of the paper, we calculate the running time and the size of the output of many operations in K in terms of the size of the input and the parameters of K. We include some earlier results about these, but we go further than them, e.g. we also analyze some R -specific operations in K like less-than comparison. In the second half of the paper, we analyze two algorithms: the Bareiss algorithm, which is an integer-preserving version of the Gaussian elimination, and the LLL algorithm, which is for lattice basis reduction. In both cases, we extend the algorithm from Z n to K n , and give a polynomial upper bound on the running time when the computations in K are performed exactly (as opposed to floating-point approximations). [ABSTRACT FROM AUTHOR]

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 non-trivial operations. The system has been successfully implemented within the Magma Computer Algebra System, and several examples are presented, using this implementation. [Copyright &y& Elsevier]