  We consider the quadratic derivative nonlinear Schr\"odinger equation (dNLS)
on the circle. In particular, we develop an infinite iteration scheme of normal
form reductions for dNLS. By combining this normal form procedure with the
Cole-Hopf transformation, we prove unconditional global well-posedness in
$L^2(\mathbb{T})$, and more generally in certain Fourier-Lebesgue spaces
$\mathcal{F} L^{s, p}(\mathbb{T})$, under the mean-zero and smallness
assumptions. As a byproduct, we construct an infinite sequence of quantities
that are invariant under the dynamics. We also show the necessity of the
smallness assumption by explicitly constructing a finite time blowup solution
with non-small mean-zero initial data.
