Rationality Groups in Prescribed Domains
Saul Epsteen · Annals of Mathematics · 1911
by the Galois' method, we first specify a domain of rationality, and a certain number of the n! permutations of the roots x1, * x**, constitute the galoisian group of the equation. Domain of rationality. A domain of rationality is a set of numbers whose sum, difference, product, and quotient (exclusive of the zero divisor) always give a number of the set. Frequently, the domain of rational numbers, r, is taken. One also uses the domain c = r[i], that is, the domain r with i = V-1 adjoined; this gives numbers of the form u + iv, u and v denoting numbers of r. Likewise, the domain of real numbers, 1?, and the domain C' = R[li], are used. In some instances, when one wvishes to state a theorem for any arbitrary domain, the symbol D is convenient. Galois' g7oV. The group G of an equation in a given domain D is the totality of substitutions having the following double property: a) Every rational function of the roots, numerically unaltered by all the substitutions of the group, has its numerical value in D; and b) Every rational function of the roots, having its numerical value in D, is numerically unaltered by the substitutions of G. In studying a linear homogeneous differential equation