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77 relations: Abelian group, Absolute value (algebra), Algebra, Algebraic geometry, Algebraic number field, Algebraic variety, American Mathematical Society, Analytic geometry, Axiom, Codomain, Commutative algebra, Complete metric space, Complex-analytic variety, Contact (mathematics), Coordinate system, Curve, Dedekind domain, Degree of a field extension, Dimension, Discrete valuation, Domain of a function, Duality (mathematics), Element (mathematics), Emil Artin, Equivalence class, Equivalence relation, Euclidean domain, Field (mathematics), Field extension, Field of fractions, Formalism (philosophy of mathematics), Function (mathematics), Geometric Algebra, Germ (mathematics), Graduate Texts in Mathematics, Group (mathematics), Group homomorphism, Group isomorphism, Image (mathematics), Index of a subgroup, Information algebra, Integer, Integral domain, Intersection number, Irreducible element, Linearly ordered group, Localization of a ring, Map (mathematics), Mathematical theory, Maximal ideal, ..., Metric (mathematics), Metric space, Multiplicative group, Multiplicity (mathematics), Ostrowski's theorem, P-adic number, P-adic order, Polynomial, Power series, Prime number, Principal ideal domain, Ramification theory of valuations, Rational number, Ring of integers, Separable extension, Set (mathematics), Spherically complete field, Springer Science+Business Media, Surjective function, Triangle inequality, Uniform space, Unique factorization domain, Valuation (measure theory), Valuation ring, W. H. Freeman and Company, Zero of a function, Zeros and poles. Expand index (27 more) »
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 (algebra)
In mathematics, an absolute value is a function which measures the "size" of elements in a field or integral domain.
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Algebra
Algebra (from Arabic "al-jabr", literally meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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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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Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry.
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American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs.
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Analytic geometry
In classical mathematics, analytic geometry, also known as coordinate geometry or Cartesian geometry, is the study of geometry using a coordinate system.
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Axiom
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
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Codomain
In mathematics, the codomain or target set of a function is the set into which all of the output of the function is constrained to fall.
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Commutative algebra
Commutative algebra is the branch of algebra that studies commutative rings, their ideals, and modules over such rings.
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Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
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Complex-analytic variety
In mathematics, specifically complex geometry, a complex-analytic variety is defined locally as the set of common zeros of finitely many analytic functions.
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Contact (mathematics)
In mathematics, two functions have a contact of order k if, at a point P, they have the same value and k equal derivatives.
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Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
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Curve
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
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Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals.
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Degree of a field extension
In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension.
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Dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it.
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Discrete valuation
In mathematics, a discrete valuation is an integer valuation on a field K; that is, a function satisfying the conditions for all x,y\in K. Note that often the trivial valuation which takes on only the values 0,\infty is explicitly excluded.
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Domain of a function
In mathematics, and more specifically in naive set theory, the domain of definition (or simply the domain) of a function is the set of "input" or argument values for which the function is defined.
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Duality (mathematics)
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.
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Element (mathematics)
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set.
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Emil Artin
Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
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Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Euclidean domain
In mathematics, more specifically in ring theory, a Euclidean domain (also called a Euclidean ring) is an integral domain that can be endowed with a Euclidean function which allows a suitable generalization of the Euclidean division of the integers.
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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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Field extension
In mathematics, and in particular, algebra, a field E is an extension field of a field F if E contains F and the operations of F are those of E restricted to F. Equivalently, F is a subfield of E. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; the real numbers are a subfield of the complex numbers.
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Field of fractions
In abstract algebra, the field of fractions of an integral domain is the smallest field in which it can be embedded.
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Formalism (philosophy of mathematics)
In foundations of mathematics, philosophy of mathematics, and philosophy of logic, formalism is a theory that holds that statements of mathematics and logic can be considered to be statements about the consequences of certain string manipulation rules.
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Function (mathematics)
In mathematics, a function was originally the idealization of how a varying quantity depends on another quantity.
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Geometric Algebra
Geometric Algebra is a book written by Emil Artin and published by Interscience Publishers, New York, in 1957.
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Germ (mathematics)
In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind which captures their shared local properties.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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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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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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Group isomorphism
In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations.
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Image (mathematics)
In mathematics, an image is the subset of a function's codomain which is the output of the function from a subset of its domain.
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Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).
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Information algebra
The term "information algebra" refers to mathematical techniques of information processing.
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Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
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Integral domain
In mathematics, and specifically in abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero.
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Intersection number
In mathematics, and especially in algebraic geometry, the intersection number generalizes the intuitive notion of counting the number of times two curves intersect to higher dimensions, multiple (more than 2) curves, and accounting properly for tangency.
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Irreducible element
In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.
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Linearly ordered group
In abstract algebra a linearly ordered or totally ordered group is a group G equipped with a total order "≤", that is translation-invariant.
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Localization of a ring
In commutative algebra, localization is a systematic method of adding multiplicative inverses to a ring.
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Map (mathematics)
In mathematics, the term mapping, sometimes shortened to map, refers to either a function, often with some sort of special structure, or a morphism in category theory, which generalizes the idea of a function.
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Mathematical theory
A mathematical theory is a subfield of mathematics that is an area of mathematical research.
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Maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals.
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Metric (mathematics)
In mathematics, a metric or distance function is a function that defines a distance between each pair of elements of a set.
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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Multiplicative group
In mathematics and group theory, the term multiplicative group refers to one of the following concepts.
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Multiplicity (mathematics)
In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset.
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Ostrowski's theorem
In number theory, Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.
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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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P-adic order
In number theory, for a given prime number, the -adic order or -adic valuation of a non-zero integer is the highest exponent such that divides.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Power series
In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant.
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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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Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element.
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Ramification theory of valuations
In mathematics, the ramification theory of valuations studies the set of extensions of a valuation v of a field K to an extension L of K. It is a generalization of the ramification theory of Dedekind domains.
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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 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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Separable extension
In field theory, a subfield of algebra, a separable extension is an algebraic field extension E\supset F such that for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial (i.e., its formal derivative is not zero; see below for other equivalent definitions).
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Set (mathematics)
In mathematics, a set is a collection of distinct objects, considered as an object in its own right.
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Spherically complete field
In mathematics, a field K with an absolute value is called spherically complete if the intersection of every decreasing sequence of balls (in the sense of the metric induced by the absolute value) is nonempty: The definition can be adapted also to a field K with a valuation v taking values in an arbitrary ordered abelian group: (K,v) is spherically complete if every collection of balls that is totally ordered by inclusion has a nonempty intersection.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Triangle inequality
In mathematics, the triangle inequality states that for any triangle, the sum of the lengths of any two sides must be greater than or equal to the length of the remaining side.
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Uniform space
In the mathematical field of topology, a uniform space is a set with a uniform structure.
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Unique factorization domain
In mathematics, a unique factorization domain (UFD) is an integral domain (a non-zero commutative ring in which the product of non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units, analogous to the fundamental theorem of arithmetic for the integers.
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Valuation (measure theory)
In measure theory, or at least in the approach to it via the domain theory, a valuation is a map from the class of open sets of a topological space to the set of positive real numbers including infinity, with certain properties.
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Valuation ring
In abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D. Given a field F, if D is a subring of F such that either x or x −1 belongs to D for every nonzero x in F, then D is said to be a valuation ring for the field F or a place of F. Since F in this case is indeed the field of fractions of D, a valuation ring for a field is a valuation ring.
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W. H. Freeman and Company
W.
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Zero of a function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
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Zeros and poles
In mathematics, a zero of a function is a value such that.
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Complete valued field, Dedekind valuation, Equivalence of valuations, Exponential valuation, Extension of a valuation, Maximal ideal of a valuation, P-adic valuation of a Dedekind domain, Prime ideal of a valuation, Ramification index of an extension of valuations, Reduced ramification index of an extension of valuations, Relative degree of an extension of valuations, Residue field of a valuation, Trivial valuation, Valuation group, Valuation ring of a valuation, Valuation theory, Value group, Valued field.
References
[1] https://en.wikipedia.org/wiki/Valuation_(algebra)
