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

130 THEORY OF EQUAT'IONS If we multiply the equations in I by o-k, ol-k, *', ton_-k, respectively, and then add, we have the more general result, o-k [W, a] = na.k III Hence, if we are given the values of the Lagrangian resolvents of an equation f(x)= 0 of the nth degree and the nth roots of unity, the equation f(x) = 0 is solved. 117. Theorem. If we operate upon the subscripts of a in [C, a] with the cyclic suibstitultion (O 1 2 3 *.. (n - 1)), [w, a] becomes w-1 [o, a]; if we operate with (0 1 2... (n - l))k, [Co, a] becomes ow [ow, a]. If we operate upon [), a] =_ a + (Oc1 + + W 01-1n with the substitution (0 1 2 *.. (n - 1)) and observe that o-l = (-1, etc., we get -1 [(0, (at] ( + ( 4a2 3 + w2a,3 + + an-l, ( + + a + + *+,.0"1-, ). Operating in this manner k times, we can easily establish the truth of the second part of the theorem. 118. Theorem. If with the cyclic substitution (O 1 2... ( - )) we operate upon the subscripts of a in [o, a], the subscript of the coefficient of each power of o in [o, a]v undergoes the cyclic substitution (0 1 2... (n- 1)),' v being any positive integer. By the Polynomial Formula expand [(, (a]- (a + a1l + ** + _lan-l)V and by the relation U" = 1 reduce all exponents of w to exponents less than n. Then combine all terms having like powers of w. We get [ow, al] = Ao + wA1 + 2A +.. + o"-ll, I

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 130
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 28, 2025.
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