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Tschirnhaus transformations
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(Definition)
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A polynomial transformation which transforms a polynomial to another with certain zero-coefficients is called a Tschirnhaus Transformation. It is thus an invertible transformation of the form
 where  are polynomials over the base field  (or some subfield of the splitting field of the polynomial being transformed). If
 then the Tschirnhaus transformation becomes a polynomial transformation mod f.
Specifically, it concerns a substitution that reduces finding the roots of the polynomial
to finding the roots of another q - with less parameters - and solving an auxiliary polynomial equation s, with
Historically, the transformation was applied to reduce the general quintic equation, to simpler resolvents. Examples due to Hermite and Klein are respectively: The principal resolvent
and the Bring-Jerrard form
Tschirnhaus transformations are also used when computing Galois groups to remove repeated roots in resolvent polynomials. Almost any transformation will work but it is extremely hard to find an efficient algorithm that can be proved to work.
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"Tschirnhaus transformations" is owned by mathcam. [ full author list (4) | owner history (1) ]
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(view preamble)
Cross-references: algorithm, Galois groups, resolvents, equation, parameters, roots, splitting field, subfield, base field, invertible, Transforms, transformation, polynomial
There are 2 references to this entry.
This is version 8 of Tschirnhaus transformations, born on 2003-08-09, modified 2005-06-29.
Object id is 4572, canonical name is TchirnhausTransformations.
Accessed 4376 times total.
Classification:
| AMS MSC: | 12E05 (Field theory and polynomials :: General field theory :: Polynomials ) |
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Pending Errata and Addenda
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![$\displaystyle \textmd{p} = T^n + a_1T^{n-1} + ... + a_n = \prod_{i=1}^n (T-r_i)\in k[T] $](http://images.planetmath.org:8080/cache/objects/4572/l2h/img5.png "$\displaystyle \textmd{p} = T^n + a_1T^{n-1} + ... + a_n = \prod_{i=1}^n (T-r_i)\in k[T] $")
