Affine Transformations of Algebraic Numbers

D. Jeffrey, Pratibha, K. Roach


We consider algebraic numbers defined by univariate polynomials over the rationals. In the syntax of \Maple, such numbers are expressed using the \texttt{RootOf} function. This paper defines a canonical form for \texttt{RootOf} with respect to affine transformations. The affine shifts of monic irreducible polynomials form a group, and the orbits of the polynomials can be used to define a canonical form. The canonical form of the polynomials then defines a canonical form for the corresponding algebraic numbers. Reducing any \texttt{RootOf} to its canonical form has the advantage that affine relations between algebraic numbers are readily identified. More generally, the reduction minimizes the number of algebraic numbers appearing in a computation, and also allows the Maple indexed \texttt{RootOf} to be used more easily.