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{{Other uses|Closure (disambiguation)}}
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{{Refimprove|date=November 2008}}
 
In [[mathematics]], particularly [[abstract algebra]], an '''algebraic closure''' of a [[field (mathematics)|field]] ''K'' is an [[algebraic extension]] of ''K'' that is [[algebraically closed field|algebraically closed]]. It is one of many [[closure (mathematics)|closures]] in mathematics.
 
Using [[Zorn's lemma]], it can be shown that every field has an algebraic closure,<ref>M. F. Atiyah and I. G. Macdonald (1969). ''Introduction to commutative algebra''. Addison-Wesley publishing Company. pp. 11-12.</ref> and that the algebraic closure of a field ''K'' is unique [[up to]] an [[isomorphism]] that [[fixed point (mathematics)|fixes]] every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''.
 
The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''.
To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''.
The algebraic closure of ''K'' is also the smallest algebraically closed field containing ''K'',
because if ''M'' is any algebraically closed field containing ''K'', then the elements of ''M'' which are [[algebraic extension|algebraic over]] ''K'' form an algebraic closure of ''K''.
 
The algebraic closure of a field ''K'' has the same [[cardinal number|cardinality]] as ''K'' if ''K'' is infinite, and is [[countably infinite]] if ''K'' is finite.
 
==Examples==
*The [[fundamental theorem of algebra]] states that the algebraic closure of the field of [[real number]]s is the field of [[complex number]]s.
 
*The algebraic closure of the field of [[rational number]]s is the field of [[algebraic number]]s.
 
*There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of '''Q'''(π).
 
*For a [[finite field]] of [[prime number|prime]] order ''p'', the algebraic closure is a [[countably infinite]] field which contains a copy of the field of order ''p''<sup>''n''</sup> for each positive [[integer]] ''n'' (and is in fact the union of these copies).<ref>{{citation|title=Infinite Algebraic Extensions of Finite Fields|volume=95|series=Contemporary Mathematics|first1=Joel V.|last1=Brawley|first2=George E.|last2=Schnibben|publisher=American Mathematical Society|year=1989|isbn=9780821854280|contribution=2.2 The Algebraic Closure of a FInite Field|pages=22–23|url=http://books.google.com/books?id=0HNfpAsMXhUC&pg=PA22}}.</ref>
 
*See also [[Puiseux expansion]].
 
==Separable closure==
An algebraic closure ''K<sup>alg</sup>'' of ''K'' contains a unique [[separable extension]] ''K<sup>sep</sup>'' of ''K'' containing all (algebraic) [[separable extension]]s of ''K'' within ''K<sup>alg</sup>''. This subextension is called a '''separable closure''' of ''K''. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of ''K<sup>sep</sup>'', of degree > 1. Saying this another way, ''K'' is contained in a ''separably-closed'' algebraic extension field. It is essentially unique ([[up to]] isomorphism).
 
The separable closure is the full algebraic closure if and only if ''K'' is a [[perfect field]]. For example, if ''K'' is a field of characteristic ''p'' and if ''X'' is transcendental over ''K'', <math>K(X)(\sqrt[p]{X}) \supset K(X)</math> is a non-separable algebraic field extension.
 
In general, the [[absolute Galois group]] of ''K'' is the Galois group of ''K<sup>sep</sup>'' over ''K''.
 
== See also==
{{portal|Mathematics}}
* [[Algebraically closed field]]
* [[Algebraic extension]]
 
==Notes==
{{reflist}}
 
{{DEFAULTSORT:Algebraic Closure}}
[[Category:Field extensions]]

Revision as of 08:36, 15 February 2014

The whole world is getting obsessed with fat reduction at when. So various people have provided in to the new 'it' thing. Slim is absolutely in. And it's today different from scenarios a few years back whenever people wish To be very thin. These days, people desire the fit and lean body.

The green coffee weight loss extract method of losing fat is 1 superior way to get rid of those extra fats. One will ask why these beans are green. It is considering these beans have not been roasted. Why the usual coffee beans most folks are familiar with are dark inside color is considering these beans have been roasted as much as 475 levels Fahrenheit. This roasting task can really make the beans lose its fat-burning and anti-oxidant element that it naturally possess. Needless to state, the green coffee fat losss are inside its most all-natural state and consequently could create people lose weight naturally.

Coffee is the world's second-most useful exported legal commodity, after oil. Exports during 2010 alone were worth certain $15.4bn, according to the International Coffee Organization.

Instead, choose fruits and vegetables for the carbohydrates. Another good choice to eat is any lean meats. Meats like poultry plus fish are better options than fatty red meats. Whenever it comes to drinks, we don't wish To drink the calories. You'll find considerable amounts of sugar plus calories in soda and juices. Make an attempt to drink lower calorie drinks or low fat milk between or with food. A terrific method to begin leading a healthier life-style is to include the assistance of all of the people in your family.

If we can lower the amount of glucose accessible from our diet, or lower the amount accessible from glycogen stores, this signifies our body is forced to burn more fat. This is what you desire whenever trying to loose fat!

Some more benefits within the green coffee beans is that it's means more affordable compared to the roasted ones. Moreover, these are natural beans plus never contain any additives. Meaning one may truly get the pure effect of chlorogenic acid found inside green coffee beans that initiates fat absorption and assists promote fat metabolism in the liver. The same acid is responsible for increasing body heat which allows the body to lose fats naturally. In addition, it also helps inside preventing the development of modern fat cells as a result of its anti-oxidant feature.

Our nation has gone wild for supplements, from capsules labeled "natural migraine relief" to medications marketed because "super-sized stress reduction." If it's sold with a label containing words like "holistic health" plus "natural," you seem to think it's secure plus efficient. The newest pumped-up pill: Green coffee bean extract for weight reduction.

Hence, utilizing these coffee beans to a minimal inside a daily diet could help you face the challenge of clearing off those calories. Including the coffee bean treatment inside your diet would just be the break you want in weight reduction.

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