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

GALOIS THEORY OF ALGEBRAIIC NUMlBERS 139 Consider the case when the product G(x) - H(x) has only integral coefficients. Then it is evident that it -1 must be divisible by m. n. Since k is relatively prime to m, and I to n, it follows that lc = pn1, 1 = 2qm, where p and q are integers. We may now write G(Sx)_ njl g1(x), H(x) q -h1(x), where the functions g1(x) and h1(x) have only integral coefficients. Consequently, if f(x) is resolvable into two rational factors G(x) and H(x), which have fractional coefficients, so that we have f()e = f (x) *. 1(x), then we have also f(x) _ y)q - /,(z) * 71(x), where the coefficients are integral throughout. Hence, if f(x) is resolvable into rational factors, it is resolvable into such factors with integral coefficients. 128. Reducibility of f(x). Whether the function f(x), in which the coefficients are integers and the degree n does not exceed 4 or 5, is reducible or not in the domain 02(), can readily be ascertained by the aid of Gauss's lelmma and ordinary algebra. We assume that, in f(x), the coefficient co, of xZ' is unity. If a0 is not unity, we can change the function so that it will be unity by taking x = -/, and multiplying by ac,"(to0 For every integral value a of x, which causes f(x) to vanish, we have a factor x- a of f(x), ~ 3. Here a must be a factor of c,. This consideration enables us always to determine the reducibility or irreducibility of functions f(x) of the second or third degree. If f(x) is of the fourth degree, then, if there is no linear rational factor, there can be no cubic rational factor. To test

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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 29, 2025.
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