In this paper we prove a sufficient condition for the existence of matchings in arbitrary groups and its linear analogue, which lead to some generalizations of the existing results in the theory of matchings in groups and central extensions of division rings. We introduce the notion of relative matchings between arrays of elements in groups and use this notion to study the behavior of matchable sets under group homomorphisms. We also present infinite families of prime numbers p such that Z/pZ does not have the acyclic matching property. Finally, we introduce the linear version of acyclic matching property and show that purely transcendental eld extensions satisfy this property.

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In this paper, we introduce the notions of matching matrices in groups and vector spaces, which lead to some necessary conditions for existence of acyclic matching in abelian groups and its linear analogue. We also study the linear local matching property in field extensions to find a dimension criterion for linear locally matchable bases. Moreover, we define the weakly locally matchable subspaces and we investigate their relations with matchable subspaces. We provide an upper bound for the dimension of primitive subspaces in a separable field extension. We employ MATLAB coding to investigate the existence of acyclic matchings in finite cyclic groups. Finally, a possible research problem on matchings in n-groups is presented. Our tools in this paper mix combinatorics and linear algebra.

The algebraic proof of the fundamental theorem of algebra uses two facts about real numbers. First, every polynomial with odd degree and real coefficients has a real root. Second, every nonnegative real number has a square root. Shipman [‘Improving the fundamental theorem of algebra’, Math. Intelligencer29(4) (2007), 9–14] showed that the assumption about odd degree polynomials is stronger than necessary; any field in which polynomials of prime degree have roots is algebraically closed. In this paper, we give a simpler proof of this result of Shipman.

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