Mathematical Areas

Basic Rings

Magma's ring hierarchy covers integers, rationals, finite fields, residue rings, and polynomial rings as fundamental structures. Arithmetic is highly optimised using specialised algorithms for each ring type.

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magma — basic-rings
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Contributors

Real and Complex Arithmetic

The complex arithmetic in Magma uses the MPC package which is being developed by Andreas Enge, Philippe Théveny and Paul Zimmermann.

Xavier Gourdon (INRIA, Paris) made available his C implementation of A. Schön-hage's splitting-circle algorithm for the fast computation of the roots of a polynomial to a specified precision. Xavier also assisted with the adaptation of his code for the Magma kernel.

Some portions of the GNU GMP multiprecision integer library are used for integer multiplication.

Most real arithmetic in Magma is based on the MPFR package which is developed by Paul Zimmermann (Nancy) and associates.

Handbook Chapter Authors

  • Introduction to Rings — W. Bosma
  • Ring of Integers — W. Bosma, A. Steel, S. Contini, B. Smith
  • Integer Residue Class Rings — W. Bosma, S. Donnelly, W. Stein
  • Rational Field — W. Bosma
  • Finite Fields — W. Bosma, A. Steel
  • Nearfields — D. Taylor
  • Univariate Polynomial Rings — A. Steel
  • Multivariate Polynomial Rings — A. Steel
  • Real and Complex Fields — W. Bosma, C. Neurohr