Solving polynomial equations with elliptic modular functions

Klein's method of solving algebraic equations is discussed and generalized to provide conditions for the unnecessity of the so-called accessory irrationality. We use the modular curve of level N to produce a "hyper-radical" of level N and discuss the accessory irrationalities involved in solving polynomial equations by means of the algebraic hypergeometric functions that define this hyper-radical. The quintic normal forms of Brioschi and Hermite elegantly fit into this framework, and we find explicit conditions for the unnecessary of accessory irrationalities for these normal forms.