7902127 FORDt DAVID JAKES ON i THE COMPUTATION OF THE MAXIMAL ORDER IK A DEDEKIND DOMAIN. THE OHIO STATE UNIVERSITY, PH.D*, 1978 University. Microfilms International 3q o n . z e e b r o a o . a n n a r b o r , mi 48ios ON THE COMPUTATION OF THE MAXIMAL ORDER IN A DEDEKIND DOMAIN DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By David James Ford, B.S., M.S. # * * K * The Ohio State University 1978 Reading Committee: Approved By Paul Ponomarev Jerome Rothstein Hans Zassenhaus Hans Zassenhaus Department of Mathematics ACKNOWLDEGMENTS I must first acknowledge the help of my Adviser, Professor Hans Zassenhaus, who was a fountain of inspiration whenever I was dry. I must also acknowlegde the help, financial and otherwise, that my parents gave me. Without it I could not have finished this work. All of the computing in this project, amounting to between one and two thousand hours of computer time, was done on a PDP-11 belonging to Drake and Ford Engineers, through the generosity of my father. VITA October 28, 1946.......... Born - Columbus, Ohio 1967...................... B.S. in Mathematics Massachusetts Institute of Technology Cambridge, Massachusetts 1967-196 8 ................. Teaching Assistant Department of Mathematics The Ohio State University Columbus, Ohio 1968-197 1................. Programmer Children's Hospital Medical Center Boston, Massachusetts 1971-1975................. Teaching Associate Department of Mathematics The Ohio State University Columbus, Ohio 1975-1978................. Programmer Prindle and Patrick, Architects Columbus, Ohio PUBLICATIONS "Relations in Q (Zh X ZQ) and Q (ZD X ZD)" (with Michael Singer) n 4 o n o o Communications in Algebra, Vol. 5, No. 1 (1977), pp. 83-87. FIELDS OF STUDY Major Field: Mathematics Studies in Algebra and Number Theory. Professor Hans Zassenhaus iii TABLE OF CONTENTS Page ACKNOWLEDGMENTS.................................................. ii VITA............................................................. iii LIST OF TABLES................................................... vi LIST OF FIGURES.................................................. vii INTRODUCTION................................ ,.................... 1 Chapter I. THE ALGEB LANGUAGE....................................... 4 0. Introduction......................................... 4 1. Preliminary Definitions............................. 4 2. Block Structure....................................... 6 3. Comments............................................. 6 4. Variable Declarations................................ 7 5. Procedure Declarations............................... 7 6. Procedure Calls...................................... 11 7. System Procedures.................................... 11 8. Input and Output Channels.................. 14 9. Integer Expressions.................................. 14 10. Boolean Expressions.................................. 15 11. Array Reference Calls................................ 16 12. Array Expressions.................................... 16 13. Destinations......................................... 17 14. Statements........................................... 18 15. Simple Statements.................................... 18 16. GOTO Statements...................................... 18 17. Assignment Statements................................ 19 18. Exchange Statements ........................... 19 19. Division Statements.......................... 19 20. Compound Statements.................................. 19 21. IF-Statements.......................... 20 22. WHILE-Statements.................................... 20 23. FOR-Statements....................................... 20 24. Labeled Statements................................... 21 iv Page II. THE ROUND TWO ALGORITHM.................................. 22 1. Background.......................................... 22 2. The Complete Algorithm.............................. 23 3. External Subroutines................................. 33 III. THE ROUND FOUR ALGORITHM................................. 36 1. Preliminary Remarks................................. 36 2. Definitions......................................... 37 3. Criteria for p-Maximality........................... 40 4. Algebraic Decomposition............................. 42 5. The p-primary case.................................. 45 IV. AN ALGEB IMPLEMENTATION OF THE ROUND FOUR ALGORITHM....... 51 1. The MAXORD Routine.................................. 51 2. Algebraic Decomposition — The DECOMP Routine....... 52 3. The p-primary Case — The NILORD Routine............ 57 4. Polynomial Factorization modulo p — The Berlekamp Algorithm............ 66 5. MINPOL — The Minimal Polynomial Routine............ 71 6. REDUCE — Hermitian Row-Reduction................... 72 7. Polynomial Manipulation Routines.................... 73 8. Routines for Manipulating Elements in a Polynomial Algebra............ 75 , 9. Miscellaneous Untyped Procedures..................... 76 10. Integer Procedures.................................. 77 V. TEST DATA AND RESULTS..................................... 79 1. 650 Polynomials..................................... 79 2. Experimental Results................................ 80 3. Conclusions.......................................... 81 BIBLIOGRAPHY................ 91 APPENDICES A. Grammar for ALGEB (Second Pass)........................... 93 B. 650 Irreducible Polynomials and Their Discriminants....... 95 v LIST OF TABLES Table 1. Execution Times LIST OF FIGURES Figure 1. Ratio of Execution Times vs. Degree INTRODUCTION If R is an integral domain and A is an overring of R, then we call an element of A integral with respect to R if it satisfies a monic poly­ nomial of R[t]. In case A is commutative, the set of all integral elements of A is an R-subring of A, called the integral closure of R in A. The integral closure of the rational integer ring Z in the complex number field C is called the ring of algebraic integers. If E is a subfield of C and at the same time a finite-dimensional extension of the rational number field Q, then the integral closure of Z in E, which .is the same as the intersection of E with the ring of algebraic integers, is called the maximal order of E. It is shown in algebraic number theory (see [2], Chapter 2) that the maximal order of E has a Z-basis w^, w,,, which is also a basis of E over Q (a so-called minimal basis). An arbitrary unital subring of E containing a basis of E over Q which is finitely generated over Z is said to be a Z-order of E. The maximal order M of E is maximal with respect to inclusion among the Z-orders of E. More generally speaking, a unital overring V of a noetherian domain R is said to be an R-order if 1„ = 1D and if V as an R-module is torsion-free V h and finitely generated and if V as a ring contains no nonzero nilpotent ideal. It is always true for any absolutely semisimple commutative hypercomplex system A over the quotient field F of R that the integral closure M of R in A is an R-order that is maximal among the R-orders of A. An R-basis of M is also an F-basis of A, also called an R-minimal basis of A over F. If A is non-commutative then there are also maximal R-orders of A but they are not necessarily uniquely determined. An example of a commutative order over a noetherian domain R is provided by the equation order Vf/R = R[t]/f(t)R[t] = R[u] corresponding to a monicseparable polynomial f of R[t]. Here we set u = t/f(t)R[t] so that the monic algebraic equation f(u) = 0 provides the defining relator for u over R. 1 2 The determination of an R-basis for M is a task that arises in many al­ gebraic settings. In the case R = Z, F = Q, and A an algebraic num­ ber field, there exists a minimal basis. Its computation provides the main task of this thesis. Knowledge of a minimal basis for an algebraic number field is of fundamental importance: it is only in the maximal order that ideals factorize uniquely as products of prime ideals. The central difficulty presents itself even in fields of degree 2. If u 2 is a root of f(x) = x - 13, A = Q[u] = Q1 + Qu, then {1,u} is a Q-basis for A. But Z[u] is not the maximal order; the element b = (u+1)/2 is 2 integral, being a root of x - x - 3. In fact {1,b} is a minimal basis for this field. In 1.927, Berwick [1] presented an algorithm to determine a minimal ba­ sis for an algebraic number field. While otherwise quite effective, this algorithm had the drawback that in certain rare cases it would fail. It was not unitl 1965, with Zassenhaus*s "First Round" algorithm [12], that we had a practical constructive method to produce a minimal basis for a number field that works in every case. The Zassenhaus al­ gorithm has the further advantage that it applies generally to any hypercomplex system with a finite basis over Q: given an order in such a hypercomplex system, the Round One algorithm produces a Z-basis for a maximal order containing the original order. The first round algorithm was improved, giving the "Second Round" algo­ rithm [14], then substantially revised for the non-coramutative case to give the "Third Round" algorithm [16]. Kehlenbach [6] programmed and tested the Round Two algorithm for the number field case. In the present work we give a new algorithm, the "Round Four" algorithm, that applies in the case of a separable polynomial algebra over Q. It is in the spirit of the Berwick method, in that it achieves its ends by manipulation of polynomials and single