Differential Galois theory

Remarks

Example: let $K = \C(x)$ with $\partial = d/dx$ and $L = K(e^x)$, the Picard–Vessiot extension of $y' = y$. An element $\sigma$ of $\Gal_\partial(L/K)$ satisfies $\partial(\sigma e^x) = \sigma(e^x)$, so $\sigma(e^x)/e^x$ has derivative $0$ and is a nonzero constant $c \in \C$. Each $c \in \C^\times$ gives such a $\sigma$, because $e^x$ is transcendental over $K$, so $\Gal_\partial(L/K) \cong \C^\times$. Non-example: let $K = \C$ with $\partial = 0$ and $L = \C(x)$ with $\partial = d/dx$; the field automorphism $\sigma$ of $L$ over $\C$ with $\sigma(x) = 2x$ satisfies clause 1 but fails clause 2, since $\partial(\sigma x) = 2$ while $\sigma(\partial x) = 1$. The elements of $\Gal_\partial(\C(x)/\C)$ are the maps $x \mapsto x + c$ with $c \in \C$, so this group is isomorphic to $(\C, +)$. For a Picard–Vessiot extension for an equation of order $n$, the group acts on the $n$-dimensional space of solutions over the constants $C$ and is a Zariski-closed subgroup of $\mathrm{GL}_n(C)$.