In many textbooks (especially those for undergraduate level), magnetic fields are described merely as a relativistic side product of electric fields when considering frames in motion relative to moving charges. Everybody knows the argument, so I will not repeat it here.
Is this view correct in terms of "deeper theory"? I learned that on university in the relativistic lectures, but nevertheless we are all talking about magnetic fields as "own entities", obeying Maxwell's equations.
Could we really remove magnetic fields from physics and still get the same laws? For my personal feeling, all we know is, that E and B are related by certain relativistic transformation rules which are intrinsically consistent, but I cannot imagine physics without B at all.
In particular, how could Faraday's law be deduced, which at first assumes no particular relative movement but makes a statement about $\partial \vec B/\partial t$ and $\nabla \times \vec E$ in fixed frames?