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Galois theory is one of the principal ways of studying such questions. 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. Extending the field ℚ \mathbb 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. 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.Ī root of an algebraic equation with integral (or equivalently, with rational) coefficients is called an algebraic number. Of course, that does not mean that addressing a problem as a Diophantine equation helps in solving it. (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. 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. Number theory (or arithmetic) studies numbers, especially the natural numbers and the integers. Weil-étale topology for arithmetic schemes

Prime field, p-adic integer, p-adic rational number, p-adic complex numberĪrithmetic geometry, function field analogy Transfinite arithmetic, cardinal arithmetic, ordinal arithmetic Peano arithmetic, second-order arithmetic natural number, integer number, rational number, real number, irrational number, complex number, quaternion, octonion, adic number, cardinal number, ordinal number, surreal number.

Higher arithmetic geometry, E-∞ arithmetic geometry