  This thesis is concerning to the Differential Galois Theory point of view of
the Supersymmetric Quantum Mechanics. The main object considered here is the
non-relativistic stationary Schr\"odinger equation, specially the integrable
cases in the sense of the Picard-Vessiot theory and the main algorithmic tools
used here are the Kovacic algorithm and the \emph{algebrization method} to
obtain linear differential equations with rational coefficients. We analyze the
Darboux transformations, Crum iterations and supersymmetric quantum mechanics
with their \emph{algebrized} versions from a Galoisian approach. Applying the
algebrization method and the Kovacic's algorithm we obtain the ground state,
the set of eigenvalues, eigenfunctions, the differential Galois groups and
eigenrings of some Schr\"odinger equation with potentials such as exactly
solvable and shape invariant potentials. Finally, we introduce one methodology
to find exactly solvable potentials: to construct other potentials, we apply
the algebrization algorithm in an inverse way since differential equations with
orthogonal polynomials and special functions as solutions.
