  We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$
with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a
lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound
$\varepsilon^{-2}$ given by the standard local theory. The proof relies on a
normal form transformation, preceded by a nonlinear transformation that
diagonalizes the operator at the highest order, which is needed because of the
quasilinear nature of the equation.
