Algebraic extensions of arbitrary integral domains

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.