Abstract
In this chapter, we turn to what today is regarded as a different branch of algebra, the solution of polynomial equations, although Gauss’s work on the ‘higher arithmetic’ was not automatically regarded as being part of algebra. We shall find that polynomial algebra also evolved in the direction of deepening conceptual insight, so here we witness again one of the origins of the transformation from school algebra to modern algebra.
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- 1.
See (Barrow-Green, Gray, and Wilson, 2018) for a recent account. In the early years, and perhaps until Gauss , it was assumed that the equations considered have real coefficients.
- 2.
See Cardano, ‘Vita Lodovici Ferarii Bonoinesis’, in Opera Omnia 9, 568–569, and Morley, H. Life of Cardan 1, 266.
- 3.
We assume for the present that these quantities do not vanish.
- 4.
See pp. 254–305 of the Oeuvres edition.
- 5.
Jordan, Traité des substitutions et des équations algébriques 1870, §387.
References
Gauss, C.F.: Demonstratio nova altera theorematis omnem functionem algebraicam rationalem integram unius variabilis in factores reales primi vel secundi gradus resolvi posse. Comm. Recentiores (Gottingae) 3, 107–142 (1816). In Werke, vol. 3, pp. 31–56
Lagrange, J.-L.: Réflexions sur la résolution algébrique des équations. Nouv. Mém de l’Académie des Sciences, Berlin, pp. 222–259 (1770/71); in Oeuvres de Lagrange 3, 205–404, J.-A. Serret (ed.) Paris
Lagrange, J.-L.: Traité de la résolution des équations numériques de tous les degrés, Paris (1st ed. 1798, 3rd ed. 1826) (1808); in Oeuvres de Lagrange 8, J.-A. Serret (ed.) Paris
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Gray, J. (2018). Is the Quintic Unsolvable?. In: A History of Abstract Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-94773-0_8
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