## 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.