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

ABELIAN EQUATIONS 217 From I we see that replacing a by f or y changes M into M, or M2. Hence, if ml = X(II) = F4 (a), MV2 = X1(IM) = F1 (a), we may write X(1I2) = F(y) = XX (XI) = F,41 (a) X1(31) = Fib(g) = X1X (11) -= F1 (a). Since, by assumption, q, l(cr) = D,4>(a), we have also XX1(M) =XX(M). Similarly for other conjugates of if. We have now proved that g(lM)= 0 is an Abelian equation. Hence we have shown that the solution of the given Abelian equation f(x) = 0 can be reduced to the solution of cyclic equations and of another Abelian equation of lower degree. The latter Abelian equation can be treated in the same manner as was f(x)= 0; hence, eventually, the solution of f(x) = 0 is reduced to that of cyclic equations only. Ex. 1. Abel gave the following example of an Abelian equation. Let a -; then cos a, cos 2 a, *.., cos na can be shown to be the roots of the equation xn x- n(n - 3) - xn-4 +.. = 0. I 4 16 1.2 For the derivation of this equation see Serret's Algebra (Ed. G. Wertheim), 1878, Vol. I, pp. 195-199. Expanding the right member of De Moivre's formula, cos ma + i sin ma = (cos a i sin a)m, by the binomial theorem, we can express cos ma as a function in Q(1) of cos a. We may, therefore, write cos ma = (cos a), where 0 is the function. Similarly, cos mna = l1(cos a). Writing mla for a in the former equation, we get cos (mm1a) -= (cos nma) = 0, (cos a). If in l(cos a) = cos mna we replace a by ma, we have cos (mnma) -= 0(cos ma) = 010(cos a). Hence every root of I can be expressed as a function in 0 of one of them, and we have in addition O0,(cos C) =- 01(cos a)o Therefore I is an Abelian equation.

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 210
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 31, 2025.
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