Abstract
Using $p$-adic methods, we shall prove the Lindemann-Weierstrass Theorem: The values taken by the exponential function at different algebraic points are linearly independent over the field of algebraic numbers. Likewise we shall prove that if $K$ is the field of rational numbers or an imaginary quadratic extension of $\mathbf{Q}$, the values at different points of $K$ of some special functions (for example the Bessel function) are linearly independent over $K$.
