Can electromagnetic fields form  tensors in a polarizable medium?

One of the bases of relativistic electromagnetism is that the electromagnetic fields \(\bf E\) and \(\bf B\) are components of an electromagnetic field tensor \(F^{\mu\nu}\). There is a simple proof of this in the absence of a polarizable medium. It has sometimes been assumed that the fields, \(\bf E\) and \(\bf B\), and their associated fields, \(\bf D\) and \(\bf H\), are also components of tensors in a linearly polarizable medium. In this paper, we question that assumption, and show that a proof that they are components of a tensor fails. Consequently, the fields \(\bf E\), \(\bf B\), \(\bf D\) and \(\bf H\) are not components of a relativistic tensor in a polarizable medium.