  The question of finding sets of monomials which are removable from a generic
homogeneous polynomial through a linear change of coordinates was raised by E.
K. Wakeford in [32]. This linear algebra question motivated C. K. Fan and J.
Losonczy to define the concept of acyclic matchings in $\mathbb{Z}^n$ in [16]
which was later generalized to abelian groups by the latter author [26].
Concepts of matchings and acyclic matchings have linear analogues developed in
the context of vector subspaces in a field extension [13, 1]. We discuss the
acyclic matching and weak acyclic matching properties and we provide results on
the existence of acyclic matchings infinite cyclic groups. As for field
extensions, we classify field extensions with the linear acyclic matching
property generalizing a theorem from [1]. The analogy between matchings in
abelian groups and in field extensions is highlighted throughout the paper and
numerous open questions are presented for further inquiry.
