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AN INTRODUCTION TO THE MODERN . THEORY OF EQUATIONS BY FLORIAN CAJOIil, Pn. D. PROFESSOR OF MATHEMATICS AT COLORADO COLLEGE Nefo gorft THE MACMILLAN COMPANY LONDON MACMILLAN CO., LTD. 1919 All right returned COPYRIGHT, BY THK MACMILLAN COMPANY. Set up and electrotypcd. Published October, 1904 J. 8. Gushing Co. Berwick Smith Co. Norwood, Mass., U. S. A. PREFACE E main difference between this text and others on the subject, published in the English language, consists in election of the material. In proceeding from the ele iry to the more advanced properties of equations, the, t of invariants and co variants is here omitted, to make for a discussion of the elements of substitutions and tuti on-groups, of domains of rationality, and of their oation to equations. Thereby the reader acquires some iarity with the fundamental results on the theory of ions, reached by Gauss, Abel, Galois, and Kronecker. e Galois theory of equations is usually found by the ner to be quite difficult of comprehension. In the pres ext the effort is made to render the subject more concrete lie insertion of numerous exercises. If, in the work of lass room, this text be found to possess any superiority, 1 be due largely to these exercises. Most of them are vn some are caken from the treatises named below. the mode of presentation T can claim no originality blowing texts have been used in the preparation of this IMANN, P. KrrhthHhinff. Leipzig, 1872. SIDE, W. Theory of Groups. Cambridge, 1897. NSIDK, W. S., and IANTON, A, W. Theory of Equations, Vol. I, 1899 Vol. II, 1901. ttsoN, L. E. Theory of Alyebraic Equations. New York, 1903, TON, B. S 77 f Const ntrtire Development of Group-Theory. Phila delphia, 1H02 yklopadieHer Mathematiachen WissenscKttften. v vi PUKFACJH GALOIS, TVifcvARisTE JSitvres mathematiques, avec une introduction par M. MILK PICARD. Paris, 1807. KLLJN, F. Vorlcfmnyen uber das Ikosaeder. Leipzig, 1884. MATTHIESSEN, L. Gnuulzwje der Antiken u. Modernen Algebra. Leip zig, 1878. NETTO, E. Theory of Substitutions, translated by F. N. COLE, Ann Arbor, 1892. NETTO, E. Vorlesuugen uber Algebra. Leipzig, Vol. I, 1890 Vol. II, 1000. PETERSEN, J. The one der Algebra ittrhen Gleichunyen Kopenhagen, 1878. PIERPONT, J. rfl oiV Theory on Alyebraic Equations Salem, 1900. SALMON, G. Modern Higher Algebra Dublin, 1876. SERKET, J. A. Handbuch der Hoheren Algebra. Deutsche Uebers. v. G. WERTHEIM. Leipzig, 1878. TODHUNTER, 1. Theory of Equations London, 1880. VOGT, II. Resolution Alyebrique des Equations. Paris, 1895. WEBER, II Lehrbueh der Algebra. Braunschweig, Vol. I, 1898 Vol. II, 1890. WEBER, H. EncyMopddie der Elementaren Algebra und Analysis. Leipzig, 190 5. Of these books, som Itave been used more than others. In the elementary parts 1 have been influenced by the excellent treatment found in the first volume of Burnside and Panton. In the presentation of the Galois theory I have followed the first volume of Webers admirable Lehrbuch der Algebra. Next to these, special mention of indebtedness is due to Bachmann, Netto, Serret, and Pier pout. I desire also to express my thanks to Miss Edith P. Hub bard, of the Cutler Academy, Miss Adelaide Denis, of the Col orado Springs High School, and Mr. R. E. Powers, of Denver, for valuable suggestions and assistance in the reading of the proofs, and to Mr. W. N. Birchby, who has furnished solutions to a large number of problems. FLORIAN CAJORI. COLORADOCOLLEGE, January, 1904. TABLE OF CONTENTS CHAPTER I PAGK SOME ELEMENTARY PROPERTIES OF EQUATIONS, 1-26 . . 1 CIIAPTEK II ELEMENTARY TRANSFORMATIONS OF EQUATIONS, 27-36 . . 31 CHAPTER III LOCATION OF THE ROOTS OF AN EQUATION, 37-61 ... 43 CHAPTER IV APPROXIMATION TO THE ROOTS OF NUMERICAL EQUATIONS, 52-58 60 CHAPTER V THE ALGEBRAIC SOLUTION OF THK CUBIC AND QDARTIC, 59-62 68 CHAPTER VI SOLUTION OF BINOMIAL EQUATIONS AND RECIPROCAL EQUATIONS, 63-67 74 CHAPTER VII, SYMMETRIC FUNCTIONS OF THE ROOTS, 68-71 ......