2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71365Using Y.André's result on differential equations staisfied by $E$-functions, we derive an improved version of the Siegel-Shidlovskii theorem. It gives a complete characterisation of algebraic relations over the algebraic numbers between values of $E$-functions at any non-zero algebraic point.9 pages 2nd version: Changed title, added reference to D.Bertrand and corrected error in definition of H in the proof of Theorem 2.5 3rd version: Added reference to vdPut/Singer, improved presentationNumber Theory11J81text