When most mathematicians use the world "algebra", they mean abstract algebra. It's not so much a research area in itself but there are many interesting subfield

When most mathematicians use the world "algebra", they mean abstract algebra. It's not so much a research area in itself but there are many interesting subfields. For example, algebraic geometry is in some sense just the study of solutions of polynomials, but there are algebraic analogues of many interesting geometric notions you encounter in plain old Riemmannian geometry. Then there is algebraic number theory, the studies of modules (generalizations of linear algebra), and then you have questions about representation theory, you can ask questions about moduli spaces of curves defined over a field, of vector bundles (or rather "coherent sheaves") defined over a field, classification of m-dimensional algebraic structure up to birational equivalence, resolution of singularities, etc. All of these have purely algebraic analogues and are topics of interest without needing to resort to the real numbers or any analytical techniques. Talking about "algebra" is a bit like talking about "analysis". It's more of a dividing line of techniques than a specific field.