Computing in Transcendental Extensions

Performing the rational operations in a field extended by a transcendental element is equivalent to performing arithmetic in the field of rational functions over the field. The computational difficulty associated with such extensions is in verifying that proposed extensions are transcendental. When the extensions being considered are functions, and where a differentiation operator can be defined for them, structure theorems can be used to determine the character of the extension and to exhibit a relationship between the adjoined element and existing quantities in case the adjoined element is not transcendental.