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| − | [[Category:Glossary]] | + | *[[Distributed Space]] ''- Research on the kind of network space required by [[the project]]'' |
| − | *Organising some notes about spacetime fundamentals from pedia | + | *[[Conceptual Space]] ''- About the space of perceived experience referred to by [[w:Taoism|Taoism]] and [[w:Advaita vedanta|Advaita vedanta]]'' |
| | + | *[[Node Space]] ''- The first attempt at designing a network space for [[the project]]'' |
| | + | *[[Hilbert space]] |
| | + | *[[Space & Time]] |
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| − | | + | == See also == |
| − | ;The fundamental theorem of algebra
| + | *[[Sacred geometry]] |
| − | In [[w:mathematics|mathematics]], the [[w:fundamental theorem of algebra|fundamental theorem of algebra]] states that every complex [[w:polynomial|polynomial]] <math>p(z)</math> in one variable and of [[w:degree of a polynomial|degree]] <math>n</math> ≥ <math>1</math> has some complex [[w:root (mathematics)|root]]. In other words, the [[w:field (mathematics)|field]] of [[w:complex number|complex number]]s is [[w:algebraically closed field|algebraically closed]] and therefore, as for any other algebraically closed field, the equation <math>p(z)=0</math> has <math>n</math> roots ([[w:multiple roots of a polynomial|not necessarily distinct]]).
| + | *[http://tetryonics101.com/ Tetryonics] |
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| − | The name of the theorem is now considered a [[w:misnomer|misnomer]] by many mathematicians, since it is an instance of [[w:mathematical analysis|analysis]] rather than [[w:algebra|algebra]].
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| − | ;Algebraically closed field
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| − | In mathematics, a [[w:field (mathematics)|field]] <math>F</math> is said to be [[w:algebraically closed|algebraically closed]] if every polynomial in one variable of degree at least <math>1</math>, with [[w:coefficient|coefficient]]s in <math>F</math>, has a zero ([[w:root (mathematics)|root]]) in <math>F</math>.
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| − | ;What is a ''field''?
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| − | In [[w:abstract algebra|abstract algebra]], a [[w:field|field]] is an [[w:algebraic structure|algebraic structure]] in which the operations of addition, subtraction, multiplication and [[w:division (mathematics)|division]] (except division by zero) may be performed, and the same rules hold which are familiar from the [[w:arithmetic|arithmetic]] of ordinary [[w:number|number]]s.
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| − | The mathematical discipline concerned with the study of fields is called [[w:field theory (mathematics)|field theory]].
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| − | ;Abstract algebra and abstract structures
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| − | [[w:Abstract algebra|Abstract algebra]] is the field of mathematics that studies algebraic structures, such as [[w:group (mathematics)|groups]], [[w:ring (mathematics)|rings]], [[w:field (mathematics)|fields]], [[w:module (mathematics)|modules]], [[w:vector space]]s, and [[w:algebra over a field|algebras]]. Most authors nowadays simply write ''algebra'' instead of ''abstract algebra''.
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| − | The term ''abstract algebra'' now refers to the study of all algebraic structures, as distinct from the [[w:elementary algebra|elementary algebra]] taught in schools, which teaches the correct rules for manipulating formulas and algebraic expressions involving [[w:real numbers|real]] and [[w:complex number|complex number]]s, and unknowns. Elementary algebra can be taken as an informal introduction to the structures known as the [[w:real field|real field]] and [[w:commutative algebra|commutative algebra]].
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| − | Contemporary mathematics and [[w:mathematical physics|mathematical physics]] make intensive use of abstract algebra; for example, theoretical physics draws on [[w:Lie algebra|Lie algebra]]s. Fields such as [[w:algebraic number theory|algebraic number theory]], [[w:algebraic topology|algebraic topology]], and [[w:algebraic geometry|algebraic geometry]] apply algebraic methods to other areas of mathematics. [[w:Representation theory|Representation theory]], roughly speaking, takes the 'abstract' out of 'abstract algebra', studying the concrete side of a given structure; see [[w:model theory|model theory]].
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| − | Two mathematical fields that study the properties of algebraic structures viewed as a whole are
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| − | [[w:universal algebra|universal algebra]] and [[w:category theory|category theory]]. Algebraic structures, together with the associated [[w:homomorphism|homomorphism]]s, form [[category theory|categories]]. Category theory is a powerful formalism for studying and comparing different algebraic structures.
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| − | In [[w:universal algebra|universal algebra]], a branch of [[w:pure mathematics|pure mathematics]], an [[w:algebraic structure|algebraic structure]] consists of one or more [[w:set|set]]s [[w:Closure (mathematics)|closed]] under one or more [[w:operations|operations]], satisfying some [[w:axiom|axiom]]s. [[w:Abstract algebra|Abstract algebra]] is primarily the study of algebraic structures and their properties.
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| − | Abstractly, an "algebraic structure," is the collection of all possible [[w:model theory|model]]s of a given set of axioms. More concretely, an algebraic structure is any particular model of some set of axioms. For example, the [[w:monster group|monster group]] both "is" an algebraic structure in the concrete sense, and abstractly, "has" the group structure in common with all other [[w:group (mathematics)|group]]s. This article employs both meanings of "structure."
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| − | This definition of an algebraic structure should not be taken as restrictive. Anything that satisfies the axioms defining a structure is an instance of that structure, regardless of how many other axioms that instance happens to have. For example, all [[w:group (mathematics)|group]]s are also [[w:semigroup|semigroup]]s and [[w:magma (algebra)|magma]]s.
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