#### Lectures in mathematics : Kyoto University

##### vol. 16

Differential algebra of nonzero characteristic / by Kotaro Okugawa
c1987

 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 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 Chapter 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 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