transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
Number theory (or arithmetic) studies numbers, especially the natural numbers and the integers. Typical questions are the distribution of prime numbers and the study of integer (or rational) solutions of algebraic equations with integer coefficients, also called Diophantine equations.
(Notice that by a theorem of Matiyasevich (spelled also Matijasevich), for every statement in mathematics (say ZF set theory) there is Diophantine equation whose solvability is equivalent to the validity of the statement. Of course, that does not mean that addressing a problem as a Diophantine equation helps in solving it. )
The analytic study of the asymptotic behaviour of the distribution of prime numbers on the positive integer line is the main subject of analytic number theory; it also studies the distribution of rational numbers with small denominators.
A root of an algebraic equation with integral (or equivalently, with rational) coefficients is called an algebraic number. Extending the field $\mathbb{Q}$ of rational numbers by abstract solutions of such an algebraic equation in a minimal way leads to an algebraic extensions of rationals, a so called number field. There are a number of algebraic structures related to the study of algebraic integers and number fields in particular; these questions comprise algebraic number theory including its central part, class field theory. Galois theory is one of the principal ways of studying such questions. Algebraic geometry is very effective in the investigation of more elaborate questions in this area, and through its historical development it has given rise to many major concepts, such as group, ideal, and Grothendieck motive.
It is one of the oldest branches of mathematics and very popular as many difficult problems can be stated in a rather elementary and simple manner, e.g. Fermat’s Last Theorem and the Riemann hypothesis.
Weil conjecture, Fermat's last problem?, abc conjecture
wikipedia en:number theory, fr:l’arithmétique modulaire
A. Fröhlich, J. W. S. Cassels (editors), Algebraic number theory, Acad. Press 1967, with many reprints; Fröhlich, Cassels, Birch, Atiyah, Wall, Gruenberg, Serre, Tate, Heilbronn, Rouqette, Kneser, Hasse, Swinerton-Dyer, Hoechsmann, systematic lecture notes from the instructional conference at Univ. of Sussex, Brighton, Sep. 1-17, 1965.
Albrecht Fröhlich, Martin J. Taylor, Algebraic number theory, Cambridge Studies in Advanced Mathematics 27, 1993
Lecture notes include
Further collection of lecture notes is here.
The following survey of Connes-Marcolli work has an accessible quick introduction to algebraic number theory