TY - JOUR
T1 - Solving polynomial equations with elliptic modular functions
AU - Schultz, Dan
N1 - Funding Information:
The author acknowledges support from National Science Foundation grant DMS-0838434.
PY - 2015/6/5
Y1 - 2015/6/5
N2 - 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.
AB - 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.
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U2 - 10.1142/S1793042115500712
DO - 10.1142/S1793042115500712
M3 - Article
AN - SCOPUS:84928813407
SN - 1793-0421
VL - 11
SP - 1313
EP - 1344
JO - International Journal of Number Theory
JF - International Journal of Number Theory
IS - 4
ER -