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t. e. Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics." [1]
The number of elements in the class group is called the class number of K. The class number of Q (√ -5 ) is 2. This means that there are only two ideal classes, the class of principal fractional ideals, and the class of a non-principal fractional ideal such as (2, 1 + √ -5 ) .
Noncommutative algebra. v. t. e. Transcendental number theory is a branch of number theory that investigates transcendental numbers (numbers that are not solutions of any polynomial equation with rational coefficients), in both qualitative and quantitative ways.
The number of ideal classes (the class number of R) may be infinite in general. In fact, every abelian group is isomorphic to the ideal class group of some Dedekind domain. [1] But if R is a ring of algebraic integers, then the class number is always finite. This is one of the main results of classical algebraic number theory.
For the real numbers used in descriptive set theory, see Baire space (set theory). In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a distance, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences.
In mathematics, a binary quadratic form is a quadratic homogeneous polynomial in two variables. where a, b, c are the coefficients. When the coefficients can be arbitrary complex numbers, most results are not specific to the case of two variables, so they are described in quadratic form. A quadratic form with integer coefficients is called an ...
Zermelo–Fraenkel set theory. In set theory, Zermelo–Fraenkel set theory, named after mathematicians Ernst Zermelo and Abraham Fraenkel, is an axiomatic system that was proposed in the early twentieth century in order to formulate a theory of sets free of paradoxes such as Russell's paradox. Today, Zermelo–Fraenkel set theory, with the ...
Noncommutative algebra. v. t. e. In mathematics, an algebraic number field (or simply number field) is an extension field of the field of rational numbers such that the field extension has finite degree (and hence is an algebraic field extension). Thus is a field that contains and has finite dimension when considered as a vector space over .
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