Abstract
Following an overview of Carl Friedrich Gauss’s Disquisitiones Arithmeticae in the previous chapter, in this chapter we turn to another major topic in Gauss’s book: cyclotomy. We will see how Gauss came to a special case of Galois theory and, in particular, to the discovery that the regular 17-sided polygon can be constructed by straight edge and circle alone.
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Notes
- 1.
Lagrange wrote \(\sqrt {-1}\) where we have i.
- 2.
Dedekind was the first to establish the irreducibility of equations of the form x φ(m) − 1 = 0, where m is not prime and φ(m) is the number of numbers relatively prime to m; see Dedekind (1857).
- 3.
See Dedekind (1873, pp. 408–409) quoted in (Shaping, 115). Article 365 will occupy us later, when we look at the reception of Wantzel’s work.
References
Dedekind, R.: Beweis für die Irreduktibilität der Kreisteilungs-Gleichungen. J. Math. 54, 27–30 (1857); in Gesammelte Mathematische Werke 1, 68–71
Dedekind, R.: Anzeige, rep. in Gesammelte Mathematische Werke 3, 408–420 (1873)
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). Cyclotomy. In: A History of Abstract Algebra. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-94773-0_5
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DOI: https://doi.org/10.1007/978-3-319-94773-0_5
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