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<p>We discuss the elegant method of Ludovico Ferrari who found a&#xa0;cubic resolvent of the quartic equation in the 16th&#xa0;century. In certain cases, his approach renders the real solutions only in terms of real quantities, which, however, involve up to four nested radicals. A&#xa0;method of Euler brings a&#xa0;reduction to only three nested radicals but makes use of complex quantities. A&#xa0;way to the said cubic resolvent that can be found in many modern textbooks turns out to be less elegant than Ferrari’s method.</p>

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Solving the quartic equation: Ferrari, Euler, and a “modern” approach

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  • 发表日期：

    2026年2月17日

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  • Kurt Girstmair

摘要

We discuss the elegant method of Ludovico Ferrari who found a cubic resolvent of the quartic equation in the 16th century. In certain cases, his approach renders the real solutions only in terms of real quantities, which, however, involve up to four nested radicals. A method of Euler brings a reduction to only three nested radicals but makes use of complex quantities. A way to the said cubic resolvent that can be found in many modern textbooks turns out to be less elegant than Ferrari’s method.