Differential ‘Galois’ extensions with new constants Extensions différentielles « galoisiennes » avec nouvelles constantes
نویسندگان
چکیده
Article history: Received 12 February 2009 Accepted after revision 6 April 2010 Presented by Bernard Malgrange Let F be a differential field with algebraically closed field of constants C and let E be a differential field extension of F . The field E is a differential Galois extension if it is generated over F by a full set of solutions of a linear homogeneous differential equation with coefficients in F and if its field of constants coincides with C. We study the differential field extensions of F that satisfy the first condition but not the second. © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Soit F un corps différentiel dont le corps des constantes C est algébriquement clos et soit E ⊃ F une extension de corps différentiels. Le corps différentiel E est une extension galoisienne différentielle de F s’il est engendré sur F par une base de solutions d’une équation différentielle linéaire homogène à coefficients dans F et si son corps des constantes est C. Nous étudions les extensions différentielles de F qui satisfont la première condition et non la seconde. © 2010 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
منابع مشابه
The Differential Galois Theory of Strongly Normal Extensions
Differential Galois theory, the theory of strongly normal extensions, has unfortunately languished. This may be due to its reliance on Kolchin’s elegant, but not widely adopted, axiomatization of the theory of algebraic groups. This paper attempts to revive the theory using a differential scheme in place of those axioms. We also avoid using a universal differential field, instead relying on a c...
متن کاملInterpretations and differential Galois extensions. DRAFT
We give model-theoretic accounts and proofs of the following results: Suppose ∂y = Ay is a linear differential equation over a differential field K of characteristic 0, and the field CK of constants of K is existentially closed in K. Then: (i) There exists a Picard-Vessiot extension L of K, namely a differential field extension L of K which is generated by a fundamental system of solutions of t...
متن کاملA History of Selected Topics in Categorical Algebra I: From Galois Theory to Abstract Commutators and Internal Groupoids
This paper is a chronological survey, with no proofs, of a direction in categorical algebra, which is based on categorical Galois theory and involves generalized central extensions, commutators, and internal groupoids in Barr exact Mal’tsev and more general categories. Galois theory proposes a notion of central extension, and motivates the study of internal groupoids, which is then used as an a...
متن کاملInaugural – Dissertation
This work presents a difference geometric approach to the strongly normal Galois theory of difference equations. In this approach, a system of ordinary difference equations is encoded in a difference extension, and the Galois groups are group schemes of finite type over the constants. The Galois groups need neither be linear nor reduced. The main result is a characterization of the extensions t...
متن کاملGeneric Picard-vessiot Extensions for Non-connected Groups
Abstract. Let K be a differential field with algebraically closed field of constants C and G a linear algebraic group over C. We provide a characterization of the K-irreducible G-torsors for nonconnected groups G in terms of the first Galois cohomology H(K, G) and use it to construct Picard-Vessiot extensions which correspond to non-trivial torsors for the infinite quaternion group, the infinit...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010