July 18, 1902.] 



SCIENCE. 



113 



reader will find as much of the obscure and un- 

 denionstrated in this attempt as has character- 

 ized other similar trials in recent years. 



Lafayette B. Mendel. 

 Sheffield Scientific School 

 OF Yale University. 



Linear Groups with an Exposition of the 

 Galois Field Theory. By L. E. Dickson, As- 

 sistant Professorof Mathematics in the Uni- 

 versity of Chicago. Leipzig, Teubner's 

 Sammlung von Lehrbuchern auf dem Ge- 

 biete der mathematischen Wissenschaften. 

 1901. Vol. VI. Pp. x-f 312. 

 In 1898 the well-known firm of B. G. Teub- 

 ner, Leipzig, Germany, began the publication 

 of the ' Encyklopadie der mathematischen 

 Wissenschaften mit Einschluss ihrer Anwen- 

 dungen,' which is intended to give in seven 

 large volumes a general outline of the known 

 parts of mathematics, together with applica- 

 tions. As a comparatively small amount of 

 space could be devoted to each subject, the 

 same firm decided to publish a large series of 

 advanced text-books in connection with this 

 encyclopedia. This series was planned espe- 

 cially to enable the authors of articles for the 

 encyclopedia to develop their subjects more 

 fully, and thus make their articles more use- 

 ful. Other writers are, however, invited to 

 make the series as complete as possible. 



More than fifty different volumes of this 

 series have already been announced by almost 

 as many different authors. The list of au- 

 thors includes a number of prominent writers 

 of various countries. The great majority of 

 these are Germans, as might have been in- 

 ferred from the fact that the work is due to 

 German influence and is published by a Ger- 

 man firm. Outside of Germany the Italians 

 seem to have been the most active collabora- 

 tors, but most of the other European coun- 

 tries, together with America, have promised 

 contributions. 



This series of text-books, together with the 

 encyclopedia, will doubtless act as a very 

 strong stimulus for greater mathematical ac- 

 tivity, and it will tend to increase in a marked 

 degree the German influence in higher mathe- 

 matics. Never before has there been such ex- 



tensive collaboration to make the recent pro- 

 gress in the various fields of mathematical re- 

 search accessible to the student. It is hoped 

 that this series will do much towards enabling 

 many additional teachers of mathematics, who 

 have suflicient leisure, to join the ranks of 

 the investigators and to assist in developing 

 the rich mines which have been opened in 

 many quarters during the past few decades. 



The present work is the sixth volume of the 

 series and is devoted to a subject which has 

 been developed principally on French and 

 American soil. The fundamental ideas are 

 due to the marvelous genius of Galois, who 

 developed them in a memoir entitled ' Sur la 

 theorie des nombres,' published in the Bulletin 

 des Sciences de M. Ferussac in 1830, when 

 their author was only eighteen years old. This 

 memoir contains the elements of a new kind 

 of imaginaries which have since been known 

 as the Galois imaginaries. They occupy prac- 

 tically the same position in the theory of con- 

 gruences as the ordinary complex numbers oc- 

 cupy in the theory of equations. 



The Galois imaginaries are generally studied 

 by means of congruences with respect to a 

 double modulus, composed of a prime number 

 p and an irreducible function of a single varia- 

 ble </' (x). It has been known for a long time 

 that the p" different residue with respect to 

 such a modulus constitute a domain of ration- 

 ality, Korper, or field. That is, if these resi- 

 dues are combined with respect to addition, 

 subtraction, multiplication or division (with 

 the exception of division by zero) the result, 

 when reduced with respect to the double mod- 

 ulus, is one of these p" residues. 



About ten years ago Professor Moore proved 

 that every finite field may be represented as a 

 field of this kind and he applied to it the iJres- 

 ent name Galois field. This important theo- 

 rem exhibits the great generality of investiga- 

 tions with respect to the Galois field. In fact, 

 with ordinary operational laws of algebra 

 the generality is complete. Imbued with the 

 beauty and interest which are attached to such 

 general investigations, the author of the pres- 

 ent volume has generalized all the systems of 

 linear groups studied by Jordan with respect 

 to the field of integers taken modulo p. He 



