Solving the Quintic: A New Approach to Bring's Transformation

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 binormal (without the 2nd, 3rd, 4th and 5th terms) form was obtained.