I don't really know how to have a clarifying discussion on such topics, but perhaps I have a little distance these days. There is such a topic as "applied algebra", though I suppose it is hardly ever called that: you break down subjects such as algebraic topology, algebraic number theory, algebraic geometry and others, by saying "what here is the algebra that is applied"? This sounds more like what is intended in the question than "group theory for group theorists", ring theory for ring theorists", and so on, counting as "pure algebra". So that Galois theory is not very relevant for algebraic topology, but is very relevant to algebraic number theory and many parts of algebraic geometry. For algebraic topology there is basic material giving access to homological algebra.

Anyway, some effort has to be made to map out areas of mathematics that are active in research terms, and to dilineate such algebra as constitutes the prerequisites, to get any relevant answers for graduate education. The approach should be global-to-local. (If it were the other way round, perhaps sheaf theory could be applied.)

I don't really know how to have a clarifying discussion on such topics, but perhaps I have a little distance these days. There is such a topic as "applied algebra", though I suppose it is hardly ever called that: you break down subjects such as algebraic topology, algebraic number theory, algebraic geometry and others, by saying "what here is the algebra that is applied"? This sounds more like what is intended in the question than "group theory for group theorists", ring theory for ring theorists", and so on, counting as "pure algebra". So that Galois theory is not very relevant for algebraic topology, but is very relevant to algebraic number theory and many parts of algebraic geometry. For algebraic topology there is basic material giving access to homological algebra.

Anyway, some effort has to be made to map out areas of mathematics that are active in research terms, and to delineate such algebra as constitutes the prerequisites, to get any relevant answers for graduate education. The approach should be global-to-local. (If it were the other way round, perhaps sheaf theory could be applied.)