Algebraic extensions of arbitrary integral domains

  • Josh D. Cohen
  • David Y. Y. Yun
4. Algebraic Fields
Part of the Lecture Notes in Computer Science book series (LNCS, volume 72)


The theory of using a polynomial to create an algebraic field extension is well-defined in mathematical literature and the technique is commonly used in symbolic computation. Even when extensions are not over a field, monics polynomials are often used to extend rings (e.g., the Gaussian Integers can be formed as ℤ[x]/x2+1). As long as only algebraic integers are introduced (i.e., the extension is monic), the computational methods are straightforward and the algorithms and supporting theory are known.

The intent of this paper is to develop the theory and algorithms necessary to understand and accommodate the use of non-monic extensions in a symbolic computing system. None of the computer algebra systems currently in operation allow such extensions. It will be shown, however, that these restrictions place an unnecessary bound on user capabilities since non-monic extension algorithms can be implemented practically and efficiently.


Computer Algebra System Algebraic Extension Greatest Common Divisor Binary Rational User Capability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 1979

Authors and Affiliations

  • Josh D. Cohen
    • 1
  • David Y. Y. Yun
    • 1
  1. 1.Mathematical Sciences DepartmentIBM Thomas J. Watson Research CenterYorktown HeightsUSA

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