<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>Since <italic>Geronimo Cardano</italic>, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same <italic>resolvente</italic>, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that <italic>Carl Friedrich Gauss</italic>’ (1799) <italic>fundamental theorem of algebra</italic> is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.</p>
</abstract>

Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We discovered two algorithms that lead to the same resolvente, each with two solutions; therefore, six formal solutions appear to solve an algebraic equation of degree four. Given that a square was utilized to derive the solution in both instances, it is critical to verify each solution. This check reveals that the four Cardanic solutions are the only four solutions to an algebraic equation of degree four. This demonstrates that Carl Friedrich Gauss’ (1799) fundamental theorem of algebra is not simple, despite the fact that it is a fundamental theorem. This seems to be a novel insight.

Solution to algebraic equations of degree 4 and the fundamental theorem of algebra by Carl Friedrich Gauss

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

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Since Geronimo Cardano, algebraic equations of degree 4 have been solved analytically. Frequently, the solution algorithm is given in its entirety. We...

Solution to algebraic equations of degree 4 and the fundamental theorem of algebra by Carl Friedrich Gauss

Published Online: Dec 08, 2022

Page range: 57 - 70

Received: Apr 05, 2022

Accepted: Sep 20, 2022

DOI: https://doi.org/10.2478/aupcsm-2022-0006

Keywords
Nonlinear algebraic equations, Niccolo Tartaglia, Johann Faulhaber, resolvente, fundamental theorem of algebra

© 2022 Norbert Südland et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Solution to algebraic equations of degree 4 and the fundamental theorem of algebra by Carl Friedrich Gauss

Published Online: Dec 08, 2022

Page range: 57 - 70

Received: Apr 05, 2022

Accepted: Sep 20, 2022

DOI: https://doi.org/10.2478/aupcsm-2022-0006

KeywordsNonlinear algebraic equations, Niccolo Tartaglia, Johann Faulhaber, resolvente, fundamental theorem of algebra

© 2022 Norbert Südland et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.

Keywords
Nonlinear algebraic equations, Niccolo Tartaglia, Johann Faulhaber, resolvente, fundamental theorem of algebra