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This paper is devoted to the so‐called Mahler method. We precisely describe the structure of linear relations between values at algebraic points of Mahler functions. Given a number field   k  , a Mahler function    f  (  z  )  ∈  k  {  z  }   , and an algebraic number   α  ,    0  <  |  α  |  <  1   , that is not a pole of   f  , we show that one can always determine whether the number    f  (  α  )    is transcendental or not. In the latter case, we further obtain that    f  (  α  )    belongs to the number field    k  (  α  )   . We also discuss some consequences of this theory concerning a classical number theoretical problem: the study of the sequence of digits of the expansion of algebraic numbers in integer bases, or, more generally in algebraic bases. Our results are derived from a recent theorem of Philippon [‘Groupes de Galois et nombres automatiques’,  J. Lond. Math. Soc . 95 (2015) 596–614] that we refine. We also simplify its proof.

Méthode de Mahler : relations linéaires, transcendance et applications aux nombres automatiques