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MODULAR INVARIANTS BY D. E. RUTHERFORD, B. Sc. CAMBRIDGE AT THE UNIVERSITY PRESS 1932 PRINTED IN GREAT BRITAIN PREFACE IN the winter of 1929 Professor Weitzenbock pointed out to me that there was no complete account of the theory of modular invariants embodying the work of Dickson, Glenn and Hazlett. The sole source of information on this subject was a number of papers, most of which ap peared in American periodicals, and a tract by Dickson which contained the substance of his Madison Colloquium Lectures. This tract, while giving a good account of the subject as it was understood in 1914, was published before the modular symbolical theory was instituted. Although the symbolical theory is not yet complete, it certainly affords a much better introduction to the subject than did the earlier non-symbolical methods. The theory is much hampered by the lack of two theorems which seem to be true but for which, as yet, no proof has been given. These are i that all congruent covariants can be represented symbolically ii that Miss Sandersons theorem can be applied to covariants as well as to invariants. In preparing the present account, the chief difficulty has been the lack of any systematic method of approach, since most of the papers on the subject have been concerned with particular cases only. My aim has been to give a clear and concise account of the theory rather than to give a complete survey of the subject, and I have therefore included in this tract only those methods which seem to be of general application. For the sake of completeness it has been necessary to include the in tricate proof of Dicksons theorem in paragraph 13. It is suggested that this might be omitted at a first reading.In order to avoid confusion the reader should notice that the words fundamental and modular vary somewhat in meaning in the different papers on the subject. I have, of course, benefited considerably from the papers of Dickson, Glenn, Hazlett, Sanderson and others, and many theorems are taken directly from their papers. The substance of Part II is largely taken from a course of lectures entitled Algebraische theorie der lichamen which Professor Weitzenbock delivered in Amsterdam University during VI PREFACE the session 1929-30. I have also made use of his lecture notes which he has kindly placed at my disposal. Professor Weitzenbock has been of great assistance to me throughout my work and has given me much helpful advice. My grateful thanks are due to Professor Turnbull of St Andrews University and to Professor Weitzenbock for reading the proof-sheets and for making many suggestions and corrections. Many thanks are also due to the Syndics of the Cambridge University Press for their helpful criticism of the manuscript. D. E. K ST ANDKEWS April 1932 CONTENTS Preface .......... page v Contents ........ r . vii SECT. PART I 1 . A new notation 1 2. Galois fields and Fermats theorem .... 1 3. Transformations in the Galois fields .... 3 4. Types of concomitants ....... 4 5. Systems and finiteness . .... 6 6. Symbolical notation 6 7. Generators of linear transformations .... 8 8. Weight and isobarism 10 9. Congruent concomitants ...... 10 10. Relation between congruent and algebraic covariants . 12 11. Formal covariants ....... 15 12. Universal covariants 15 13. Dicksons theorem ....... 17 14. Formal invariants of the linear form . . .22 15. The use of symbolical operators 24 16. Annihilatorsof formal invariants . .... 26 17. Dicksons method for formal covariants ... 28 18. Symbolical representation of pseudo-isobaric formal Co variants ......... 30 19. Classes 31 20. Characteristic invariants 33 21. Syzygies 35 22. Residual covariants 36 23. Miss Sandersons theorem 39 24. A method of finding characteristic invariants . . 42 25. Smallest full systems 43 26 Residual invariants of linear forms .... 45 VU1 CONTENTS BECT. 27. Residual invariants of quadratic forms. . . . page 47 28. Cubic and higher forms 51 29...
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