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


  • Norbert Südland Aage Gmbh, Germany
  • Jörg Volkmann DSI-NRF Centre of Excellence in Mathematical and Statistical Sciences, School of Computer Science and Applied Mathematics, University of the Witwatersrand, Johannesburg, South Africa
  • Dinesh Kumar College of Agriculture-Jodhpur, Agriculture University Jodhpur, Jodhpur, India


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


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.


Bronstein, I. N. and K. A. Semendjajew. Taschenbuch der Mathematik. Moscow: Nauka, Leipzig: BSB B. G. Teubner Verlagsgesellschaft, 1987.

Duc Goninaz, Michel and Claude Roux, ed.Plena ilustrita vortaro de Esperanto. Paris: Sennacieca Asocio Tutmonda, 2005.

Gellert W. et al. Kleine Enzyklopädie: Mathematik. Leipzig: VEB Bibliographisches Institut, 1974.

Hawlitschek, Kurt. Johann Faulhaber 1580–1635, Eine Blütezeit der mathematischen Wissenschaften in Ulm. Stadtbibliothek Ulm, 1995.

Südland, Norbert, and Armin Kadow. "Solvo de algebraj ekvacioj". Last modified November 21, 2015.




How to Cite

Südland, N., Volkmann, J., & Kumar, D. (2022). Solution to algebraic equations of degree~4 and the fundamental theorem of algebra by Carl Friedrich Gauss. Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, 21, 57–70. Retrieved from