This somewhat suppresses new questions and seems unnecessary, but what strikes me is: Field and Galois theory is hardly ring theory in most situations, certainly not in the ones named in the examples. One could just as well tag these group-theory. Tagging them galois-extensions would be much more justifiable. So I think this is actually cluttering the ring-theory tag.
Don’t we have policies against possibly counterproductive retag sprees?
Evil suspicion: Does retagging old questions one has answered add to the specific tag badge process?