In this thesis we will introduce the Differential Galois Theory, also known as the Picard-Vessiot Theory, which takes up the ideas of the classical Galois Theory and applies them to the study of the solutions of differential equations. We will define the Picard-Vessiot extensions for homogeneous linear differential equations and their associated differential Galois groups and then prove the Fundamental Theorem of the Differential Galois Theory. In conclusion there will be a brief discussion of Liouville extensions and their generalization with the aim of proving a solvability criterion by generalized quadratures of a differential equation. The tools necessary for the understanding of this theory are presented in the thesis. In fact, notions of Differential Algebra will be introduced (derivations, rings, fields and differential extensions, etc...) and useful concepts will be presented, such as the tensor product (of differential rings), affine varieties and algebraic groups, together with their properties.
In questa tesi verrà introdotta la Teoria di Galois Differenziale, detta anche Teoria di Picard-Vessiot, la quale riprende le idee della Teoria di Galois classica e le applica allo studio delle soluzioni di equazioni differenziali. Verranno definite le estensioni di Picard-Vessiot per equazioni differenziali lineari omogenee e i relativi gruppi di Galois differenziali associati, per poi dimostrare il Teorema Fondamentale della Teoria di Galois Differenziale. In conclusione vi sarà una breve trattazione delle estensioni di Liouville e della loro generalizzazione con l'obiettivo di dimostrare un criterio di risolubilità per quadrature generalizzate di una equazione differenziale. Gli strumenti necessari alla comprensione di tale teoria sono esposti all'interno della tesi. Verranno infatti introdotte nozioni di Algebra Differenziale (derivazioni, anelli, campi ed estensioni differenziali, ecc...) e concetti utili come il prodotto tensoriale (tra anelli differenziali), le varietà affini ed i gruppi algebrici, insieme alle loro proprietà.
Una introduzione alla Teoria di Galois Differenziale
TACCHETTI, EDOARDO
2022/2023
Abstract
In this thesis we will introduce the Differential Galois Theory, also known as the Picard-Vessiot Theory, which takes up the ideas of the classical Galois Theory and applies them to the study of the solutions of differential equations. We will define the Picard-Vessiot extensions for homogeneous linear differential equations and their associated differential Galois groups and then prove the Fundamental Theorem of the Differential Galois Theory. In conclusion there will be a brief discussion of Liouville extensions and their generalization with the aim of proving a solvability criterion by generalized quadratures of a differential equation. The tools necessary for the understanding of this theory are presented in the thesis. In fact, notions of Differential Algebra will be introduced (derivations, rings, fields and differential extensions, etc...) and useful concepts will be presented, such as the tensor product (of differential rings), affine varieties and algebraic groups, together with their properties.File | Dimensione | Formato | |
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https://hdl.handle.net/20.500.12608/52230