Algebra is the discipline of pure mathematics that is concerned with the study of the abstract properties of a set, once this is endowed with one or more operations that respect certain rules (axioms) ...

To restore it, we need Dedekind's concept of ideals. However, one still needs the supporting concepts of algebraic number field and algebraic integer, and the supporting theory of rings, vector spaces ...

Algebraic geometry deals primarily with solutions lying in an algebraically closed field, while arithmetic geometry deals with the more subtle study of solutions lying in a number field or its rings ...