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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"?

Victor
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    "I cannot distinguish algebra from number theory". See also, Analytic Number Theory. – Arkady Sep 09 '16 at 17:15
  • Thanks! Well, I feel analytic number theory is defined by the methodology or tools used in number theory. Of course we can use analytic method in number theory. But my question focuses on what objects are studied by number theory. – Victor Sep 09 '16 at 17:23
  • Something that for historical, cultural, emotional, bureaucratic and sometimes even logical reasons we call number theory – Lorenzo Sep 09 '16 at 19:07
  • studying the behavior of certain weird sequences of integers or rational numbers (the first of them being the prime numbers), and also studying the same weird sequences in more complicated rings than $\mathbb{Z}$ and $\mathbb{Q}$ : in number fields or in polynomial rings – reuns Sep 09 '16 at 23:10
  • Also posted to MO, http://mathoverflow.net/questions/249449/what-is-exactly-number-theory – Gerry Myerson Sep 10 '16 at 05:49

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Of course all fields overlap, but roughly speaking number theory studies the structure of numbers. Analysis studies the structure of functions. (Are they continuous, differentiable, bounded, etc.) Algebra studies the structure of sets under unary and binary operations. (Is this $+$ commutative, associative, is the kernal of this map closed under $+$?) Number theory cares about the numbers themselves. If you have 6 blocks, you can arrange them into a $2\times 3$ rectangle or a $1\times 6$ rectangle. But if you have 7 blocks, you can form only one rectangle. So the numbers $6$ and $7$ have different personalities, and we study those differences (and samenesses) and use what we learn to solve problems.

Also, as you say, it's not just the integers. Irrational numbers have different personalities too. Some are "regular" and some might be 3-order approximable (or whatever they call it.) Some are "algebraic" and some are "transcendental".

B. Goddard
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