Algebraic number theory and Integer

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Difference between Algebraic number theory and Integer

Algebraic number theory vs. Integer

Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

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Absolute value

In mathematics, the absolute value or modulus of a real number is the non-negative value of without regard to its sign.

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Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

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Algebraic integer

In algebraic number theory, an algebraic integer is a complex number that is a root of some monic polynomial (a polynomial whose leading coefficient is 1) with coefficients in (the set of integers).

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Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

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Euclidean algorithm

. EXAMPLES CAN BE FOUND BELOW, E.G., IN THE "Matrix method" SECTION.

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Field (mathematics)

In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.

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Fundamental theorem of arithmetic

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that, moreover, this representation is unique, up to (except for) the order of the factors.

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Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

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Latin

Latin (Latin: lingua latīna) is a classical language belonging to the Italic branch of the Indo-European languages.

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Modular arithmetic

In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value—the modulus (plural moduli).

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P-adic number

In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems.

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Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

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Ring (mathematics)

In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra.

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Ring of integers

In mathematics, the ring of integers of an algebraic number field is the ring of all integral elements contained in.

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