  In this paper, we investigate the existence of an elementary abelian closure
in characteristic not $2$ for biquadratic extensions. We discover that it
exists for any non-cyclic extension. We make use of it to obtain a
classification for this class of extensions up to isomorphism via descent. This
permits us to describe the geometry of this moduli space in group theoretic
terms. We also provide two families of polynomials with two parameters that can
describe any quartic extensions.
