Abstract algebra and Archimedean property

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Difference between Abstract algebra and Archimedean property

Abstract algebra vs. Archimedean property

In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are sets with specific operations acting on their elements. In abstract algebra and analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, is a property held by some algebraic structures, such as ordered or normed groups, and fields.

Algebraic structure

In mathematics, an algebraic structure consists of a nonempty set A (called the underlying set, carrier set or domain), a collection of operations on A (typically binary operations such as addition and multiplication), and a finite set of identities, known as axioms, that these operations must satisfy.

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Characteristic (algebra)

In mathematics, the characteristic of a ring, often denoted, is defined to be the smallest positive number of copies of the ring's multiplicative identity that will sum to the additive identity.

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David Hilbert

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of his time.

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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 the corresponding operations on rational and real numbers.

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Group (mathematics)

In mathematics, a group is a set with an operation that associates an element of the set to every pair of elements of the set (as does every binary operation) and satisfies the following constraints: the operation is associative, it has an identity element, and every element of the set has an inverse element.

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Monoid

In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element.

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P-adic number

In number theory, given a prime number, the -adic numbers form an extension of the rational numbers which is distinct from the real numbers, though with some similar properties; -adic numbers can be written in a form similar to (possibly infinite) decimals, but with digits based on a prime number rather than ten, and extending to the left rather than to the right.

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Polynomial

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication and exponentiation to nonnegative integer powers, and has a finite number of terms.

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Real number

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.

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Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist.

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