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2018 | OriginalPaper | Buchkapitel

10. Galois’s Theory

verfasst von : Jeremy Gray

Erschienen in: A History of Abstract Algebra

Verlag: Springer International Publishing

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Abstract

This chapter focuses on Galois’s work. Can it be that although he had done his best to present the complete resolution of the question “When is a polynomial equation solvable by radicals”, what we can see are pieces of this resolution, key pieces presumably, but we don’t find them convincing? Is it that the pieces are not all there, or is there something about them that we cannot see? There is a way of thinking about the problem that Galois presented which we do not follow. Is it because he was too quick? In that case, patient work should enable us to catch up. Or is there some deep insight he has failed to present?

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Vorheriges Kapitel The Unsolvability of the Quintic

Nächstes Kapitel After Galois

There are many accounts, not all of which agree on even the most basic facts; this account follows (Rothman 1989). For an analysis of how Galois has been portrayed in fiction, and his transformation into a cultural icon (which was very evident in Paris in 2012) see (Weber and Albrecht 2011).

At this time political assemblies were banned, so dinners were a popular, and legal, substitute.

Quoted in (Rothman 1989, pp. 165–166).

Raspail lived to receive the Cross of the Legion of Honour from Louis-Philippe, and was all his life a militant republican. A boulevard and a metro stop in Paris are named after him.

Quoted in (Rothman 1989, p. 94).

Galois’s memoir is available in English in Neumann, The Mathematical Writings of Évariste Galois (2011), and also in Edwards, Galois Theory .

All these results are taken from (Neumann 2011, pp. 111–113).

What has to be shown is that x ₁ is the only root of the two equations, see (Edwards 1984, pp. 44–45).

Taken from (Neumann 2011, pp. 113–115).

Taken from (Neumann 2011, p. 113).

Edwards (1984, p. 54) explains that an “algebraic equation” is one with letters rather than numbers as its coefficients.

This example comes from (Bolza 1893, p. 99), which is Bolza’s review of the English translation of (Netto 1882), see Sect. 15.

Taken from (Neumann 2011, p. 119).

(Neumann 2011, p. 121), emphasis Galois’s.

Some of these connections are also traced in (Bottazzini and Gray 2013).

Bolza, O.: Review of Netto, E. The theory of substitutions and its applications to algebra (transl. F.N. Cole, 1892). Bull. Am. Math. Soc. 2, 83–106 (1893)

Bottazzini, U., Gray J.J.: Hidden Harmony – Geometric Fantasies: The Rise of Complex Function Theory. Springer, New York (2013)CrossRef

Boucard, J.: Louis Poinsot et la théorie de l’ordre: un chaînon manquant entre Gauss et Galois? Rev. Hist. Math. 17, 41–138 (2011)MathSciNetMATH

Brechenmacher, F.: Self-portraits with Évariste Galois (and the shadow of Camille Jordan). Rev. Hist. Math. 17, 273–372 (2011)MathSciNetMATH

Edwards, H.M.: Galois Theory. Springer, New York (1984)MATH

Ehrhardt, C.: Évariste Galois and the social time of mathematics. Rev. Hist. Math. 17, 175–210 (2011)MathSciNetMATH

Galois, É.: Sur la théorie des nombres. Bull. Sci. Math. Phys. Chim. 13, 428–435 (1830); rep. in (Galois 1846, 398–405) and in (Neumann 2011, 61–75)

Goldstein, C.: Charles Hermite’s stroll through the Galois fields. Rev. Hist. Math. 17, 211–272 (2011)

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

Netto, E.: Substitutionentheorie und ihre Anwendungen auf die Algebra. Teubner, Leipzig (1882); English transl. The Theory of Substitutions and its Applications to Algebra. P.N. Cole (transl.) The Register Publishing Company, 1892

Neumann, P.M.: The concept of primitivity in group theory and the second memoir of Galois. Arch. Hist. Exact Sci. 60, 379–429 (2006)MathSciNetCrossRef

Neumann, P.M.: The Mathematical Writings of Évariste Galois. Heritage of European Mathematics. European Mathematical Society, Zürich (2011)

Rothman, T.: Science à la Mode. Princeton University Press, Princeton (1989)

Titel: Galois’s Theory
verfasst von: Jeremy Gray
Verlag: Springer International Publishing
Buch: A History of Abstract Algebra
Print ISBN: 978-3-319-94772-3

Electronic ISBN: 978-3-319-94773-0

Copyright-Jahr: 2018
DOI: https://doi.org/10.1007/978-3-319-94773-0_10

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