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September 6th, 2012, 06:11 PM   #101
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Re: Solvable Quintics

Quote:
 Originally Posted by mathbalarka I want to share something about solving general quintic and the Kiepert's algorithm, are you interested?
most definitely yes.

 September 6th, 2012, 11:03 PM #102 Math Team     Joined: Mar 2012 From: India, West Bengal Posts: 3,871 Thanks: 86 Math Focus: Number Theory Re: Solvable Quintics The most useful algorithmic process of solving a general quintic is Kiepert's algorithm. Here's the flow chart for it : $\text{General Quintic} \rightarrow \text{Principal Quintic} \rightarrow \text{Brioschi Normal Form} \rightarrow \text{Jacobi Sextic} \rightarrow \text{Elliptic Functions}$ Whereas, another algorithm is : $\text{General Quintic} \rightarrow \text{Principal Quintic} \rightarrow \text{Bring-Jerrard Normal Form} \rightarrow \text{Bring Radical}$ But the part where we convert the principal quintic in the Bring-Jerrard form, we use a quartic Tschrinhausen transformation which is rather harder than just transforming Principal form into the Brioschi normal form. The next stage of the Kiepert algorithm transforms the Brioschi normal form into the Jacobi sextic form. This can be very beneficial, since a sextic has 6 roots. Hence now we have the same group with a permutation of six elements. Now O. Perron shows that all the roots can be derived in terms of elliptic curves. Now appropriate trivial transformations gives back the original general quintic with the 5 roots (6-th root has been discarded). Balarka .

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