Im pretty familiar with single variable calculus, basic linear algebra, and a spattering of basic discrete math topics and proof techniques. I feel like I at least have a concept of what most more advanced math topics cover. Like multivariable and vector calculus extending calculus ideas to higher dimensions, differential equations and PDE’s I can make sense of their basic purpose and idea, real analysis makes sense, but abstract algebra just seems so weird and foreign in comparison.
Looking at basic definitions of the subject I can sort of follow along with some of the basic definitions, group, ring, etc. But I am really just at a loss as to exactly what the overarching idea is. Or how exactly it relates to the “algebra” learned in high school. I feel like I could really use a more broad overview of exactly what the subject entails, what it sets out to explore, and what kinds of problems it was developed to solve.
Any further reading on the history of the subject or any good overviews (that would be understandble to a first year student in mathematics) are also appreciated.