CEU Electronic Theses and Dissertations, 2018
| Author | Nagy, Levente |
|---|---|
| Title | A General Theory of Solution Algebras in Differential and Difference Galois Theory |
| Summary | In a recent paper, Yves André introduced a refinement of the differential Galois correspondence over algebraically closed constant fields k of characteristic zero, where he characterized closed subgroups of the differential Galois group corresponding to intermediate extensions (called solution fields) generated by some but not necessarily all solutions. Moreover, he showed that these solution fields are fraction fields of of so-called solution algebras and obtained a Galois correspondence between these solution algebras and affine quasi-homogeneous varieties under the differential Galois group. In the present thesis, first we generalize Picard-Vessiot theory to an abstract, categorical setting, proving a bijection between Picard-Vessiot rings and fibre functors. Using this abstract theory, we also prove the above mentioned correspondence between solution algebras and affine algebraic quasi-homogeneous schemes in this more general setting. We then discuss iterative differential rings and difference rings, in each case we develop the theory of solution fields and using previously known results, we establish the connection between solution fields and observable subgroups of the Galois group scheme. |
| Supervisor | Szamuely, Tamás |
| Department | Mathematics PhD |
| Full text | https://www.etd.ceu.edu/2018/nagy_levente.pdf |
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