Maths Project Repository Database

Galois theory studies solutions to polynomial equations by looking at how various roots of a polynomial are related to each other. More precisely, the structure of the field extensions obtained by adjoining roots of a polynomial are studied via the group of symmetries of the roots. This provides a link between field theory and group theory, which culminates in Galois theorem establishing a bijective correspondence between intermediate fields of a Galois field extension $K$ and subgroups of the Galois group $\text{Gal}(L/K)$.

Differential Galois theory (or Picard-Vessiot theory) uses similar ideas to study solutions to homogenous linear differential equations. An ordinary field is now replaced by a field $K$ equipped with a notion of differentiation, and the goal is to study the space of solutions of a differential equation (in a larger differential field $L$ containing $K$, the "Picard-Vessiot extension"). The differential Galois theorem establishes a bijective correspondence between intermediate differential fields of a Picard-Vessiot extension $K$ and closed subgroups of its differential Galois group $\text{dGal}(L/K)$.