Abstract
E30 in the Enestrom index. Translated from the Latin original "De formis
radicum aequationum cuiusque ordinis coniectatio" (1733). For an equation of
degree n, Euler wants to define a "resolvent equation" of degree n-1 whose
roots are related to the roots of the original equation. Thus by solving the
resolvent we can solve the original equation. In sections 2 to 7 he works this
out for quadratic, cubic and biquadratic equations. Apparently he gives a new
method for solving the quartic in section 5. Then in section 8 Euler says that
he wants to try the same approach for solving the quintic equation and general
nth degree equations. In the rest of the paper Euler tries to figure out in
what cases resolvents will work.
Two references I found useful were Chapter 14, p.p. 106-113 of C. Edward
Sandifer, "The Early Mathematics of Leonhard Euler", published 2007 by The
Mathematical Association of America and Olaf Neumann, "Cyclotomy: from Euler
through Vandermonde to Gauss", p.p. 323-362 in the collection "Leonhard Euler:
Life, Work and Legacy" edited by Bradley and Sandifer, 2007. Stacy Langton has
given a lot of details about Euler's work on the theory of equations, and also
some advice on the translation; of course any mistakes are my own. If Langton
ends up writing anything about Euler' and the theory of equations I would
highly recommend reading it.
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