Algebraicity of the Appell–Lauricella and Horn hypergeometric functions

We extend Schwarzʼ list of irreducible algebraic Gauss functions to the four classes of Appell–Lauricella functions in several variables and the 14 complete Horn functions in two variables. This gives an example of a family of functions such that for any number of variables there are infinitely many algebraic functions, namely the Lauricella FC functions.