Computational Problems, Methods, and Results in Algebraic Number Theory
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262 Horst G. Zimmer Mathematisches Institut II der Universitat Karlsruhe (TH), Karlsruhe/DEUTSCH LAND
Computational Problems, Methods, and Results in Algebraic Number Theory
Springer-Verlag Berlin· Heidelberg· NewYork 1972
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
262 Horst G. Zimmer Mathematisches Institut II der Universitat Karlsruhe (TH), Karlsruhe/DEUTSCH LAND
Computational Problems, Methods, and Results in Algebraic Number Theory
Springer-Verlag Berlin· Heidelberg· NewYork 1972
AMS Subject Classifications (1970): 12-02, 12-04, 12A99, 12B99, 12C99, 12D99, 12F99; 14-02, 14-04, 14G99, 14H99, 14)99, 14K99
ISBN 3-540-05822-2 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-05822-2 Springer-Verlag New York· Heidelberg· Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-78191. Printed in Germany.
Offsetdruck: Julius Beltz, Hernsbach/Bergstr.
Preface
In this report an attempt is being made to briefly survey, with special emphasis on the computer-oriented view point, numerical investigations centering on Algebraic Number Theory. The report was initiated while the author attended the Atlas Symposium No. 2 on 'Computers in NUmber Theory' that toOk place at Oxford, England, August 18-23, 1969.
It was written mainly during the author's participation in the Algebra
Year 1969/70 at the University of California at Los Angeles.
The author wishes
to express his thanks to the Atlas Computer Laboratory at Chilton, Didcot, and to the Department of Mathematics of the University of California at Los Angeles for inviting him to these meetings. Thanks are also due to the referee for some corrections and additional references and to Mrs. Elaine Stafford of the Mathematics Department of the University of California at Los Angeles for kindly taking care of the typing of the manuscript.
Karlsruhe, November 1971
Horst G. Zimmer
Contents
Introduction • • 1. Finite Fields
2
2. Factorization of Polynomials.
5
3. Galois Groups
11
4. Continued Fractions
14
5. Field Extensions • . . • • • (a) Primitive Elements • • • • • . (b) Tschirnhausen Transformations (c) A Real Root Calculus.
18 18
19 22
6. Modules and Orders • • • • • (a) Sum and Intersection of Modules •• (b) Embedding of an Order into a Maximal Order •• (c) The Fractional Ideals of an Order
24 24
7. Products of Linear Forms.
29
8. Units in Algebraic Number Fields.
35
9. Class Numbers of Algebraic Number Fields.
40
25 27
10. Class Groups a
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