Alternatively, after undergoing a Tschirnhausen transformation, the related icosahedral equation can be used to find a solution in terms of hypergeometric functions.  
Algebraic surfaces of this type include the Togliatti surface and equations of this type include the de Moivre equation. These equations may be solved by reducing them to the Bring form and applying Jacobi theta functions.  Alternatively, after undergoing a Tschirnhausen transformation, the related icosahedral equation can be used to find a solution in terms of hypergeometric functions.  Those and other methods are now known, but there can never be a general equation for them using only field and radical operations.  FTP, name this class of algebraic equations whose study led to the development of Galois groups and Abel's impossibility theorem.
Algebraic surfaces of this type include the Togliatti surface and equations of this type include the de Moivre equation.
quintic (or fifth-degree or other clear equivalents)