September 6th, 2012, 06:11 PM  #101  
Math Team Joined: Jul 2011 From: North America, 42nd parallel Posts: 3,372 Thanks: 233  Re: Solvable Quintics Quote:
 
September 6th, 2012, 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 : 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 . 

Tags 
quintics, solvable 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Tschirnhaus transformation and quintics  Surgborg  Abstract Algebra  28  August 4th, 2013 07:40 AM 
Advanced Analysis of Quintics  mathbalarka  Algebra  6  March 29th, 2013 02:00 PM 
Quintics : Easy!  mathbalarka  Algebra  6  January 24th, 2013 01:03 AM 
Quintics  mathbalarka  Algebra  8  September 29th, 2012 08:19 AM 
Algorithmic Process For Solving Quintics?  mathbalarka  Algebra  6  September 2nd, 2012 12:05 PM 