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Introduction to Theory of Quintics


Well-known member
MHB Math Helper
Mar 22, 2013
This is a graph containing the main Tschirnhausen transformations of a general quintic and the algorithms to solve them. The methods would be explicitly described most probably elsewhere in this forum


  • The lower-most nodes are the elementariest functions to which the root of a general quintic can be extracted.
  • In any sub-graph of the above, the lowermost elements are the form reduced from the topmost elements via Tschirnhausen transformations.
  • Deeper colors indicates the effectiveness of the algorithm in computational research, as well as popularity.
  • Dotted lines indicates a reduction that is non-obvious although not a part of the interest, in most cases.
  • Black lines indicates a transformation that is not part of the algorithm.
  • The red arrows indicate the Kiepert algorithm, which I like to call Kiepert-Perron algorithm.
  • The green arrows indicate Glasser's derivation.
  • The magenta-like colors indicate Hasner's method and Kronecker-Brioschi derivation respectively.

Any comment/question regarding this thread should be posted in http://mathhelpboards.com/commentary-threads-53/commentary-introduction-theory-quintics-8210.html
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