On maximal and minimal linear matching property

The matching basis in field extentions is introduced by S. Eliahou and C. Lecouvey in [2]. In this paper we define the minimal and maximal linear matching property for field extensions and prove that if $K$ is not algebraically closed, then $K$ has minimal linear matching property. In this paper we will prove that algebraic number fields have maximal linear matching property. We also give a shorter proof of a result established in [6] on the fundamental theorem of algebra.