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

CHAPTER XVII CYCLIC EQUATIONS 170. Definition. A cyclic equation is one whose Galois group is the cyclic group, ~ 101. Kronecker called such equations "einfache Abel'sche Gleichulmgen." A quadratic equatiorn is, cyclic; for the Galois group is the symmetric group G2(, which is at the same time the cyclic group of the second degree. The general cubic is not a cyclic equation in the domain defined by its coefficients; for its Galois group is G(), which is not a cyclic group. However, if we adjoin VD =- (a - R1)2a 1- f2)(a1 - a2), the Galois group becomes (~ 163) G(3, which is cyclic. Hence the general cubic is cyclic in the domain ~(a,a, a,,VD)) The general qtuartic is not a cyclic equation in the domain defined by its coefficients, but if we adjoin a function which belongs to the cyclic group G4~)I, the equation is cyclic in the new domain. One such function that may be adjoined is M = 3ac12 + (1(92 + t2 + a a2. If n is a prime number, xn-l + xn-2 +... + aX + 1 = I is a cyclic equation in the domain Q(). For, ~ 130, this equation is irreducible. The cyclic function 012W2 + 2 + s. + W. +,^_-2W is seen by the relations o2 - o12, 03= o13, etc., to be equal to the sum of the roots, which is - 1. Therefore the Galois 187

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Title
An introduction to the modern theory of equations, by Florian Cajori.
Author
Cajori, Florian, 1859-1930.
Canvas
Page 170
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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