Algebraic extensions of arbitrary integral domains

Abstract

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]/x ²+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.