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Title Differential Algebra of Nonzero Characteristic Author(S) Title Differential algebra of nonzero characteristic Author(s) Okugawa, Kôtaro Citation Lectures in Mathematics (1987), 16 Issue Date 1987 URL http://hdl.handle.net/2433/84921 Right Type Book Textversion publisher Kyoto University LECTURES MATHEMATICS Department Mathematics KYOTO UNIVERSITY Differential Algebra of Nonzero Characteristic BY Kôtaro. OKUGAWA Publ ished by KINOKUNIYA CO., Ltd. Tokyo, Japan 16IN LECTURES IN MATHEMATICS Department of Mathematics KYOTO UNIVERSITY 16 Differential Algebra of Nonzero Characteristic By KOtaro OKUGAWA Published by KINOKUNIYA CO ., Ltd. copyright 1987 by KINOKUNIYA Co., Ltd. ALL RIGHT RESERVED Printed in Japan Differential Algebra of Nonzero Characteristic By K6taro OKUGAWA Acknowledgment The author is deeply grateful to Department of Mathematics of Kyoto University which accepted this note to publish in "Lectures in Mathematics , Kyoto University". Contents page Foreword ix Chapter 1 Derivations 1.1 Conventions 1 1.2 Definitions and elementary properties 2 1.3 Examples of derivations 7 1.4 Derivatives of powers 12 1.5 Taylor expansion 13 1.6 Rings of quotients 15 1.7 Separably algebraic extension fields 20 1.8 Inseparably algebraic extension fields 24 Chapter 2 Differential Rings and Differential Fields 2.1 Definitions 31 2.2 Differential ring of quotients 35 2.3 Differential polynomials 37 2.4 Differential ideals 44 2.5 Differential homomorphisms and differential isomorphisms 48 2.6 Contractions and extensions of differential ideals 49 2.7 Separably algebraic extension fields of a differential field 54 2.8 Inseparably algebraic extension fields of a differential field 55 2.9 The field of constants of a differential field 58 -v- page Chapter 3 Differential Ideals 3.1 Perfect and prime differential ideals 61 3.2 Conditions of Noether 65 3.3 Differential rings satisfying the condition of Noether for ideals 67 3.4 Differential polynomial rings 70 3.5 Linear differential polynomials 74 3.6 Linear dependence over constants 77 3.7 Results about constants 82 3.8 Extension of the differential field of coefficients 85 Ch apt er 4 Universal Differential Extension Field 4.1 Definitions 95 4.2 Lemmas 96 4.3 The existence theorem 99 4.4 Some properties of the universal differential extension field 100 4.5 Linear homogeneous differential polynomial ideals 101 4.6 Primitive elements 102 4.7 Exponential elements 104 4.8 Weierstrassian elements 107 Chapter 5 Strongly Normal Extensions 5.1 Some properties of differential closure 112 5.2 Conventions 116 5.3 Differential isomorphisms 116 5.4 Specializations of differential isomorphisms 119 -v1 - page 5.5 Strong differential isomorphisms 125 5.6 Strongly normal extensions and Galois groups 133 5.7 The fundamental theorems 141 5.8 Examples 152 5.9 Differential Galois cohomology 154 Chapter 6 Picard-Vessiot Extensions 6.1 Picard-Vessiot extension whose Galois group is the general linear group 160 6.2 Fundamental theorems of Galois theory for Picard-Vessiot extensions 163 6.3 Picard-Vessiot extension by a primitive 168 6.4 Picard-Vessiot extension by an exponential 171 6.5 Liouvillian extensions 173 Bibliography 185 Index of Notations 187 Index of Terminologies 189 -vii- Foreword The theory of Galois type on differential fields of charac- teristic zero was established by E. R. Kolchin. Namely, the theory of Picard-Vessiot extensions was developed in [1], [2] and others, and then, more generally, the theory of strongly normal extensions was presented in [3] and others. On the latter theory, some contributions were made by H. Matsumura, C. Chevalley and S. Lang. We can now find a fine treatise on such materials in the book [4] of Kolchin. Galois groups of Picard-Vessiot extensions are algebraic matric groups, and those of strongly normal extensions are algebraic groups. Since general algebraic groups are well stu- died also in the case of nonzero characteristic, it is interest- ing to develop the corresponding theory of Golois type on dif- ferential fields of nonzero characteristic. To do this, it seems appropriate to deal with differential fields and rings which are defined by associating a set of mutually commutative iterative higher derivations of infinite rank. From this stand- point, we provided in [5] a method of developing the theory of Picard-Vessiot extensions of arbitrary characteristic. Since then, basic properties of such differential fields and rings of nonzero characteristic have been more elaborately considered by us; recently, we had contributions [7], [8], [10] and [11] of K. Tsuji (née Shikishima). Although complicated calculations and observations are necessary, useful results are obtained, and -IX- the Galois theory of strongly normal extensions of differential field of nonzero characteristic is established. This note contains, in the case of nonzero characteristic p, detailed descriptions of basic properties of differential fields and rings and of strongly normal extensions of differen- tial fields. In preparing the note, we use freely the notions and results of Chap.O and Chap.V of [4] and Chap.I of [1], because these chapters deal with basic algebraic notions includ- ing that of the algebraic group for arbitrary characteristic and they are very fundamental literatures. Thus materials from these chapters are often introduced as known ones to the reader, although we endeavor to cite the origin of each of them. Gene- rally speaking, we tried to use terminologies and notations of the whole book [4] which is excellent and well-equipped. Our definition of "derivation" in Chap.l leads to formal difference between each differential algebraic concept of this note and the corresponding one of [4], but a same term or a same nota- tion is used to each pair of corresponding concepts. On the other hand, a number of new differential algebraic concepts and notations must be introduced in this note. Chap.l gives fundamental considerations on "derivation". Chap.2 and Chap.3 deal with basic properties of differential fields, differential rings and differential ideals. Chap.4 is devoted maimly to the proof of existence of universal differen- tial extension field of any given differential field. Chap.5 --x- contains the Galois theory of strongly normal extensions of dif- ferential fields. This chapter is developed following Chap.VI of [4], although the concern about separability necessitates additional discussions on many steps. In Chap.6, some obser- vations are made on Picard-Vessiot extensions and Liouvillian extensions of differential fields. This shows that the theory of [5] can be rearranged more naturally under the new light of this note. Theorems are numbered consecutively from beginning to end of each chapter; so are propositions and lemmas. In quoting a result, we indicate not only the chapter but also the section; examples: Th.2 of §1.7; part (a) of Th.10 of §2.6; Cor.2 to Th.1 of §3.1. However, in referring to a result within the same section, we do not mention the section. We wish to thank heartily E. R. Kolchin, the presence of whose works initiated us into our study and helped us. We ex- press with sincere thanks that many results (§1.8, §2.8, §4.3, §4.8, §5.1, §5.6, §5.7, Examp.3 of §5.8) of this note are ori- ginally due to K. Tsuji, and that many helpful criticisms on the preparatory manuscript have been given by our colleagues (of Kyoto Sangyo Univ.) and by our friends. March 1987 KOtaro Okugawa CHAPTER 1 Derivations 1.1. Conventions The term field is used for commutative field of nonzero characteristic p arbitrarily fixed once for all except for some examples. The term ring is used for commutative unitary ring which contains some field as a unitary subring. If a ring and its subring are considered, the latter is tacitly supposed to be a unitary subring of the former. If a ring- homomorphism is considered, it is also tacitly supposed uni- tary. If X1, X2, ... are to appear as subscripts, superscripts or exponents, we use instead X(1), X(2), ... respectively in order to simplify the typewiting. This convention is never applied to the characteristic p, and p(m) denotes always the power pm for any m E Z (Z being the set of integers). N denotes the set of natural number (including 0). Let I be a finite or infinite set. Modifying the notation for the set NI, we use the notation N(I)for the set of all families (v) = (v(i)( i E I) whose components v(i) (i E I) are in N and zeros except for a finite number. The family (0) means .N (I). thefamily has(v) witha v(i)= 0 for all i E I canonical structure of an additive semigroup having (0) as . When the set I is finite,N (I). zerothe element coincides with N • -1- 1.2. Definitions and elementary properties Let R be a ring. We mean by a derivation S of R an iterative higher derivation of infinite rank, that is, an infi- nite sequencesequE S = (6vi v E N) of mappings(5 v of R into R which satisfiessat the following conditions (cf. [5]): (D1)o= o = idR (the identity mapping of R), (D2)6(x 6 v(x + y) = 6vx + 6vy, (D3) Sv(xy (xy) =v(1)+v(2)=v6v(1)x-6v(2)17, (D4) SX6sx=(A+P)(s Ax+vx for all ,y E R and for all X,p,v E N. We say that (Sv is the member of order v of S or the with member of S (v E 6 N). Since v(v E Ni are mappings of R into R, we can make productprodu (that is, composite) of any finite number of them; we met alralready the product of 6A and 6 1.1in (D4).
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