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J. Buhler, Z. Reichstein, * On Tschirnhaus transformations*,
in Topics in Number Theory, edited by
S. D. Ahlgren et. al., Kluwer Academic Publishers, pp. 127-142, 1999.

** Abstract: **
We revisit the classical problem of simplifying
polynomials by means of Tschirnhaus transformations. We consider
Tschirnhaus transformations involving (i) no auxiliary radicals,
(ii) arbitrary radicals, (iii) odd degree radicals, and (iv) odd degree
radicals and the square root of the discriminant. We previously
showed that by using substitutions of type (i) one cannot reduce
the general polynomial of degree n to a form with less than [n/2]
independent coefficients. In this paper we give a new proof of this
result and also extend it to transformations of types (iii) and (iv).
In the last section we present alternative proofs, based
on the cohomological approach shown to us by J.-P. Serre.

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