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Modern Algebra is also called abstract algebra at some schools. As the name implies, it deals with very abstract concepts. For example, instead of working with real numbers, we generalize operations to a set of objects called a group, in which the elements are closed under an "addition" operation (that is, the "addition" of any two elements in the set produces another element in the set), there exists an identity element, and for each element in the group there exists a "negative" of that element. After defining sets of objects like groups, rings, and fields, you then learn how to prove dozens of theorems and corollaries about them. Modern algebra and real analysis (in which you prove all the theorems from calculus) are very rigorous classes, and I wouldn't recommend them right after multivariable calculus. Modern algebra has a few esoteric applications, such as theoretical physics, and it's usually only taken by math majors. I think you should look at differential equations, since that's what most people take after multivariable and linear algebra. DiffEq is also applicable to a wide variety of fields, such as biology, chemistry, physics, and engineering.
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