# Introduction

The Galois theory of linear differential equations, or differential Galois theory, is the analogue of the classical Galois theory of polynomials for linear differential equations. Generally speaking one studies linear differential equations, that is differential equations of the form

$$
L(y)=a_n y^{(n)}+a_{n-1} y^{(n-1)}+\cdots+a_1 y^{(1)}+ a_0 y= 0,
$$

in which the coefficients $a_i$ are contained in some ring. The natural analogue of a field in the classical case is the notion of a differential field, that is a specific case of a differential ring. A *differential ring* $F$ is equipped with an additive map $\delta_F:F \to F$ called a *derivation*, satisfying the multiplicative rule

$$
\delta_F(a\cdot b)=\delta_F(a)\cdot b + a\cdot\delta_F(b),
    \quad a,b\in F.
$$

A classical derivation is the usual derivative. All differential rings have a ring structure and have a map defined on them. A differential ring that is also a field is called a *differential field*.

The differential rings have type `RngDiff` and their elements have type `RngDiffElt`. All differential rings contain a *differential ring of constants* on which the derivation acts as the zero map. The differential rings and their elements inherit all functionality of the rings from which the differential ring is created. We call the ring from which a differential ring $F$ is created the *underlying ring* of $F$.

A solution of a differential equation is an element of some differential field. It can happen that a solution is not an element of a given differential field $F$, but is an element of a differential extension of $F$. By this we mean a differential field (ring) $M$ with $F\subset M$ such that the derivations satisfy $\delta_M|_F=\delta_F$. This is completely analogous to field extensions induced by solutions of a polynomial.

To clearly describe linear differential equations in Magma we formalize the concept of taking the derivative. To a differential field $F$ with derivation $\delta_F$, one associates a non–commutative ring $F[D]$, *the ring of linear differential operators*. An element of $F[D]$ is called a *differential operator*. A differential operator of *degree* $n\in{\mathbb{Z}}_{\ge0}$ in $F[D]$ is of the form

$$
L=a_nD^n+a_{n-1}D^{n-1}+\cdots+a_1D+a_0,
$$

with $a_n \not=0$ and all $a_i\in F$. Addition in $F[D]$ is term–wise and the multiplication of elements in $F[D]$ is determined by the rule

$$
D*a=aD+\delta_F(a), \quad a\in F.
$$

With these concepts $L(y)=0$ is the linear differential equation

$$
a_n \delta_F^n(y)+a_{n-1} \delta_F^{n-1}(y)+
     \cdots+a_1 \delta_F(y)+a_0 y= 0.
$$

For an introduction to the basic concepts in differential Galois theory, one is encouraged to consult [[van der Put and Singer, 2003](../../references.md#cite-vdps03)]. This book is used as the basis for the implementation of differential rings, fields and operator rings in Magma.