Galois Groups of Second and Third Order Linear Differential Equations

Using the representation theory of groups, we are able to give simple necessary and suucient conditions regarding the structure of the galois groups of second and third order linear diierential equations. These allow us to give simple necessary and suucient conditions for a second order linear diierential equation to have liouvillian solutions and for a third order linear diierential equation to have liouvillian solutions or be solvable in terms of second order equations. In many cases these conditions also allow us to determine the group. 1. Introduction Let k be a diierential eld yy with algebraically closed eld of constants C and L(y) = 0 a linear diierential equation zz with coeecients in this eld. One can form the m th