Abstract
Bulletin of the American Mathematical Society, February.— The opening articles respectively give abstracts of the proceedings and papers read at the sixth annual meeting, at New York, December 28, 1899, by Prof. F. N. Cole, and at the sixth semi-annual meeting, at Chicago, December 28 and 29, 1899, by Prof. T. F. Holgate.—On cyclical quartic surfaces in space of n dimensions, by Dr. V. Snyder, was read at the first of the above meetings. The method employed is a generalisation of that first employed by Darboux, using Lie's more general co-ordinates. For n = 2 (bicircular quartic curves) reference is made to memoirs by Casey, Darboux, Cox, Loria and others, where the curves have been discussed from a different point of view, and for n = 3 (cyclides) reference is again made to Casey, and to Maxwell, Cayley, Darboux, Reye, Loria, Bôcher and others. In the case of n = 4, the number of distinct types is 58, and of n = higher numbers, the number of types has not been determined.—At the same meeting, Prof. H. Taber read a paper on the singular transformations of groups generated by infinitesimal transformations, and Prof. Dickson gave a proof of the existence of the Galois field of order pr for every integer r and prime number p. Existence proofs have been given by Serret (Alg. Sup. vol. 2) and by Jordan (Traité des Substit. pp. 16, 17). The developments used by Serret are lengthy, and the short proof by Jordan assumes with Galois the existence of imaginary roots of an irreducible congruence modulo p. The present proof proceeds by induction. Assuming the existence of the GF[pn], it derives that of the GF[pnq], q being an arbitrary prime number. Since the GF[p] exists, being the field of integers taken modulo p, it follows that the GF[pq] exists, and by a simple induction that the GF[pr] exists for r arbitrary.—Dr. Lovett contributes a lengthy review of the “Leçons nouvelies sur l'analyse infinitésimale et ses applications géométriques” of Ch. Méray (1st part, 1894; 2nd part, 1895; 3rd Part, 1897; and 4th part, 1898).—Varied information of interest to mathematicians occupies the “Notes” and “New Publications.”
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Scientific Serials . Nature 61, 554–555 (1900). https://doi.org/10.1038/061554b0
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DOI: https://doi.org/10.1038/061554b0