Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Solution of the Quintic and Sextic by Radicals

Version 1 : Received: 8 November 2021 / Approved: 9 November 2021 / Online: 9 November 2021 (13:59:06 CET)

How to cite: Franco, J.A. Solution of the Quintic and Sextic by Radicals. Preprints 2021, 2021110178 (doi: 10.20944/preprints202111.0178.v1). Franco, J.A. Solution of the Quintic and Sextic by Radicals. Preprints 2021, 2021110178 (doi: 10.20944/preprints202111.0178.v1).

Abstract

Applying a procedure similar to that of E.S. Bring, by using a 4th degree Tschirnhaus transformation, it was possible to transform the Bring-Jerrard normal quintic (BJQ) equation into a De Moivre form (DMQ), so that it could be solved by radicals. The general solution by radicals of the De Moivre equations of any degree is presented. By the same procedure the BJSx (normal sextic) equation was taken to another one without the 2nd, 4th and 6th terms which was transformed into a cubic (solvable) equation. By applying a 6th degree Tschirnhaus transformation to the BJSp (normal septic) equation its Bring-Jerrard binormal (without the 2nd, 3rd, 4th and 5th terms) form was obtained.

Keywords

Solution of the Quintic by Radicals; Solution of the Sextic by Radicals; Solution of the De Moivre Equation of any degree by Radicals; Eliminating Four Terms at once from the Septic.

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