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.
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory