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Introduction à la Thrie des Nombres Algèbriques

Nature volume 88, page 443 (01 February 1912) | Download Citation



THIS book is a French translation of the work on algebraic numbers by Dr. J. Sommer, professor of the Technical High School of Danzig. It is an elementary exposition designed to be an introduction to the researches of modern German mathematicians, particularly of David Hubert, whose masterly rèsumè in “Jahrsbericht der deutschen Mathematiker Vereinigung, 1895,” is likely to be regarded as a classic. Gauss was the first to enlarge the field of the higher arithmetic by including therein numbers of the form a+b√–1. This led to a beautiful theory in the development of which he was followed by Dirichlet, Kummer, Dedekind, and Kronecker, to mention only a few of the most important and successful investigators. The results included extensions and generalisations of many theorems of the higher arithmetic, and in particular of the theorem of quadratic reciprocity. The modern theory of algebraic numbers involves a further extension of the domain of numbers in that every root of an irreducible algebraic equation with rational coefficients is said to be an algebraic number. For a given degree of such an equation the corpus of such numbers comprises every rational function of such roots. When n is 2 we have the quadratic corpus which involves the irrationality √m, m being a given integer, not a perfect square, which defines the corpus. This book treats the domain of these numbers with some completeness (pp. 16–183).

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