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Most of the number theory textbooks I've dealt with take a very classical approach to the subject. I'm looking for a textbook that's something like a first course in number theory for people who have a decent command of modern algebra at the level of something like Lang's Algebra. Does such a book exist, and if it does, what is it called?
In the introduction to Ireland and Rosen, they note something that was bugging me for a while, "Nevertheless it is remarkable how a modicum of group and ring theory introduces unexpected order into the subject. This is precisely the perspective I was looking for, so if anyone passes by this topic looking for a book that approaches number theory in this way, I feel like this quote should point him her?
Well, it depends on the actual subject you want to approach and the "decent command of modern algebra" already assumed; without knowing more, I would recommend:.
Number fields by Marcus. Just as the title says, a great!
Elementary Number Theory -- from Wolfram MathWorld
Academic Press. It explains the basics class field theory, zeta functions to understand the Langlands Program. A course on arithmetic by Serre. Graduate Texts in Mathematics. P-adic fields, quadratic forms, zeta functions and modular forms.
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If I understand it well now, what you want are books about basic number theory with a good algebraic foundation. I can recommend the following:. Elementary methods in number theory , by Nathanson.
Elementary Number Theory
It starts low, but it reaches quite high. Modern number theory is a broad subject that is classified into subheadings such as elementary number theory, algebraic number theory, analytic number theory, geometric number theory, and probabilistic number theory. These categories reflect the methods used to address problems concerning the integers.
The ability to count dates back to prehistoric times. This is evident from archaeological artifacts , such as a 10,year-old bone from the Congo region of Africa with tally marks scratched upon it—signs of an unknown ancestor counting something.
It is certain that an understanding of numbers existed in ancient Mesopotamia, Egypt , China, and India, for tablets, papyri, and temple carvings from these early cultures have survived. A Babylonian tablet known as Plimpton c. This certainly reveals a degree of number theoretic sophistication in ancient Babylon.
Despite such isolated results, a general theory of numbers was nonexistent. According to tradition, Pythagoras c. His philosophy enshrined number as the unifying concept necessary for understanding everything from planetary motion to musical harmony.
Given this viewpoint, it is not surprising that the Pythagoreans attributed quasi-rational properties to certain numbers.