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- Subjects studied in number theory 3 answers
To be specific, I have a solid background in college level math but I feel that I have trouble to understand what is number theory. As a first approximation, number theory studies integers. But it is of course not a complete answer. Somebody told me number theory in general studies "arithmetical properties" in algebraic systems. But this is not an answer for me but merely a paraphrase, since number theory=arithmetic. So what exactly number theory?
Let me put my question in another way, I cannot distinguish algebra from number theory.
Let me take geometry and topology as an example. These two areas are of course highly correlated. But I think I understand the difference: when we talk about geometry, the manifold should be equipped with metric. And curvature is extremely important in geometry. My understanding may be shallow here. But at least I find some key features like "metric", "curvature" geometrical but not topological.
But what's going on for number theory? I will greatly appreciate that someone can give a one sentence answer to distinguish number theory from algebra.
Edit: This question was marked as a duplicate of the question: Subjects studied in number theory
They are similar but I think we are asking a question from different levels. I'm familiar with the basic number theory research subject such as prime distribution, zeta function, diophantine equation, etc.
But for the higher level number theory especially algebraic number theory, I fail to understand the key difference between number theory and general algebra.
For example, let me quote class field theory from wikipedia: "In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields and function fields of curves over finite fields and arithmetic properties of such abelian extensions." After reading this, I will say uh-oh, why abelian extensions is number theory? Wouldn't it be algebra? What properties would be considered as "arithmetic property"?