September 6th, 2012, 05:11 PM  #101  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Solvable Quintics Quote:
 
September 6th, 2012, 10:03 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 : Whereas, another algorithm is : But the part where we convert the principal quintic in the BringJerrard 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 (6th root has been discarded). Balarka . 

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quintics, solvable 
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