An introduction to the modern theory of equations, by Florian Cajori.

CYCLIC EQUATIONS 203 an equation of the first degree and second degree. Hence there is a one-to-one correspondence between constructions by ruler and compasses and algebraic operations which are purely rational or involve square roots. Consequently, if we wish to show the impossibility of constructing a quantity by ruler and compasses, we need only show that the algebraic expression for that quantity in terms of the known quantities cannot be given by a finite number of square roots. Applying these ideas to the problem of dividing the circle into n equal parts by means of ruler and compasses, the problem is possible or impossible according as the roots of n_ - 1 = 0 can be expressed by a finite number of square roots or not. If n is a prime number of the form 2k + 1, the degree n - 1 of the cyclotomic equation is a power of 2, and the operations called for in ~ 182 involve square roots only. Hence, when n is a prime of the form 2 + 1, the divisioa of the circle into n equal parts by r nler tad compasses is always possible. This important result is due to Gauss. Ex. 1. Solve x5 — 1 = 0 by Gauss's method. The cyclotomic equation is x4+x3+x2++1=-0. Here n-1=4=2 2; e = 2, f = 2. It is only necessary to solve two quadratics. By trial we get for n = 5, g = 2 the roots, cog, OWg', Cgg3 these yield the two periods 71 =t + Wg2 = + W 4, 71- = Cq + C- s3 = C2 -+ c3. Hence equation IV, ~ 180, becomes X2 (7- 7+ 1) + 71 = 0. But + 711 = -C + 02 + c3 + 4 =-, and ra71 = (co + WI) (W2 + co3)= W3 + 3 2 + W + w4 = 1. Hence the quadratic takes the form 2 + X - =, and x - 2

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 190
Publication
New York,: The Macmillan company,
1904.
Subject terms
Equations, Theory of
Group theory.

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"An introduction to the modern theory of equations, by Florian Cajori." In the digital collection University of Michigan Historical Math Collection. https://name.umdl.umich.edu/abv2146.0001.001. University of Michigan Library Digital Collections. Accessed May 29, 2025.
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