On the inverse problem of Galois theory of differential fields
On the inverse problem of Galois theory of differential fields
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关于伽罗瓦微分场理论的反问题
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发表时间：
1964
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通讯作者：
A. Białynicki
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作者：
A. Białynicki
1.这里考虑的所有场的特性都是0。设F是域，C是F的代数闭子域.设G是定义在C上的连通代数群。F（G）表示G上定义在F上的所有有理函数的域。如果gCG，则F（g）表示g在F上生成的场。我们将说F（G）的导子与G*（C）可换，如果它与g* 可换，对于每个gEG（C），其中g* 表示由g的左平移引起的F（G）的自同构，即，（g*f）（x）=f（gx），对于任意xCG。F表示F（G）的所有导子在F上为零且与G*（F）可换的李代数。若G_1是定义在F上的G的正规子群，则F（G/G_1）与F（G）的一个子域正则同构;我们将确定F（G/G_1）和这个子域。如果R是整环，则（R）表示R的分数域。R的每个导子d都可以唯一地扩展为R的一个导子（扩展的导子也用d表示）。如果F1、F2是包含F作为子场的两个场，并且如果d1、d2分别是F1、F2的导数，使得d1 i F= d2 I F并且d1（F）C F，则d1 i d2表示对于每个aGF 1和bCF 2，由（d1 d2）（a 0 B）-d1（a）Gb+ a0 d2（B）确定的F1、0 F F2的导数。do表示一个域的零导数（我们头脑中的域总是很清楚的）。常微分场53的基础场将由F表示。
1. All fields considered here are of characteristic 0. Let F be a field, let C be an algebraically closed subfield of F. Let G be a connected algebraic group defined over C. F(G) denotes the field of all rational functions on G defined over F. If gCG then F(g) denotes the field generated by g over F. We shall say that a derivation of F(G) commutes with G*(C) if it commutes with g*, for every gEG(C), where g* denotes the automorphism of F(G) induced by the left translation by g, i.e., (g*f)(x) =f(gx), for any xCG. F denotes the Lie algebra of all derivations of F(G) that are zero on F and which commute with G*(F). If G1 is a normal subgroup of G defined over F then F(G/G1) is canonically isomorphic to a subfield of F(G); we shall identify F(G/G1) and this subfield. If R is an integral domain then (R) denotes the field of fractions of R. Every derivation d of R can be uniquely extended to a derivation of R (the extended derivation will be also denoted by d). If F1, F2 are two fields containing F as a subfield and if d1, d2 are derivations of F1, F2, respectively, such that d1i F= d2 I F and d1(F) C F then d1i d2 denotes the derivation of F1,0F F2 determined by (d1 d2)(a 0 b) -d1(a)Gb+a0d2(b), for every aGF1 and bCF2. do denotes the zero derivation of a field (it will be always clear what field we have in mind). The underlying field of an ordinary differential field 53 will be denoted by F.