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2014-02-10T14:58:22Z

Disambiguate Reliability to wikt:reliability using popups; formatting: 15x whitespace, 4x heading-style, nbsp-dash (using Advisor.js)

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	~~In [[field theory (mathematics)\|field theory]], the '''Stufe''' (/[[Help:IPA for German\|ʃtuːfə]]/; German: level) <math>s(F)</math>~~ of a ~~[[field (mathematics)\|field]]~~ <~~math~~>F<~~/math~~> ~~is the least number of squares that sum~~ to ~~<math>-1</math>~~. If <~~math~~>-1<~~/math~~> ~~cannot~~ be ~~written as~~ a ~~sum of squares~~, ~~<math>s(F)=\infty</math>~~. ~~In this case~~, ~~<math>F</math> is~~ a ~~[[formally real field]]~~. ~~[[Albrecht Pfister (mathematician)\|Albrecht Pfister]] proved~~ that ~~the Stufe, if finite, is always a power~~ of ~~2, and that conversely every power~~ of ~~2 occurs~~.~~<ref name=R13/>~~		Do not allow food to be your only associated with enjoyment. Find relief . center their lives around cooking and eating. And there is nothing wrong with this. Food is often tons of fun. You should ensure possess other stuff you enjoy quite food. Find a new hobby, especially person that requires movements.<br><br>You will want to try out because you've got nothing to lose and such to realize. If you've been dreaming about burning fat the simplest way, without BS claims and prospects, then this is it for a person will.<br><br>In can be a big phase, you need to join a gym. No exceptions. If you want get associated with fat quickly, you have to join a gym. Exercise 3 five days every week. You need to focus on mild workouts that do not involve an excessive amount of strength. The weights make use of must not be too heavy. You need to burn a few of your calories via task. Make sure you do not spend more than a quarter-hour on physical fitness.<br><br>[http://xbound.ncity.net/xe/?document_srl=240173 raspberry trim Review] [http://www.vitalyoga.org/ForumRetrieve.aspx?ForumID=3229&TopicID=616318&NoTemplate=False raspberry trim Reviews] But will be the major sources wherein you is designed for the extra weight issues. Wait a second, from all these matter one can't judge that carbohydrates is think again required by our body we still need that will. There's a reason for everything. Ok! Depriving from carbohydrates might bring lots of changes affecting the different organs among the body can cause condition. So, in order to not to achieve to such stage there were a hottest weight loss product within the market nowadays partner.e. Beyond [http://www.buildfoxvalley.com/ForumRetrieve.aspx?ForumID=317&TopicID=824447&NoTemplate=False raspberry trim] ketone.<br><br>The chemical compound within raspberries delivers it its wonderful smell and tang. This chemical is most notably known as, Rubus Ideaeus. The chemical is most widely known for its flavoring of foods and beverages for many people years. Stressed recent years, this chemical has been looked at for advantages in weight. Studies proven that the ketones stimulate a hormone in shape called Apdiponectin to burn excess fat for energy and make the immune system; and they completely tried and true.<br><br>To stay motivated while losing weight, reward yourself in methods aren't associated food. As small treat from time to time can be good, but try to leave out of the habit of smoking of pondering over food as a reward. Instead, promise yourself a completely new outfit--in great deal higher size!--or a vacation in the hot spa tub.<br><br>But lots of lies the actual fact which you cannot decide yourself if you wish to which fat loss methods will need use exactly what is good mix a variety of techniques at this point perfect with regard to you. So have to consult someone expert, result in overheating a diet expert, physician, your mentor or a person. Let someone professional do are for the person.<br><br>The capacity to reduce weight and preserve it off for that long term takes the right state of mind. When you decide that losing weight is in order to you, your discipline will certainly make it easier for of which you do the crucial modifications necessary you to get rid of your excess fat.

	~~== Powers~~ of ~~2 ==~~

	~~If <math>s(F)\ne\infty~~<~~/math~~> ~~then~~ <~~math~~>~~s(F)=2^k<~~/~~math> for some <math>k\in\Bbb N<~~/~~math>~~.~~<ref name=R13>Rajwade (1993) p~~.~~13<~~/~~ref><ref name~~=~~Lam379>Lam (2005) p.379</ref>~~

	~~''Proof~~:~~'' Let <math>k \in \Bbb N<~~/~~math> be chosen such that <math>2^k \leq s(F) < 2^{k+1}<~~/~~math>~~. ~~Let <math>n = 2^k<~~/~~math>~~. ~~Then there are <math>s~~ = ~~s(F)</math> elements <math>e_1, \ldots, e_s \in F\setminus\{0\}</math> such that~~

	~~:<math>0~~ = ~~\underbrace{1 + e_1^2 + \cdots + e_{n-1}^2 }_{=: a} + \underbrace{e_n^2 + \cdots + e_s^2}_{~~=~~: b}\;. </math>~~

	~~Both <math>a</math> and <math>b</math> are sums of <math>n</math> squares, and <math>a \ne 0</math>, since otherwise <math>s(F)\le 2^k</math>, contrary to the assumption on <math>k</math>.~~

	~~According to the theory of [[Pfister form~~]~~]s,~~ the ~~product <math>ab</math>~~ is ~~itself~~ a ~~sum of <math>n</math> squares~~, that is~~, <math>ab = c_1^2 + \cdots + c_n^2</math>~~ for ~~some <math>c_i \in F</math>~~. ~~But since <math>a+b=0</math>~~, ~~we also have <math>-~~a~~^2 = ab<~~/~~math>, and hence~~

	~~:<math>-1 = \frac{ab}{a^2} = \left(\frac{c_1}{a} \right)^2 + \cdots +\left(\frac{c_n}{a} \right)^2\;,<~~/~~math>~~

	~~and thus <math>s(F) = n = 2^k<~~/~~math>~~.

	== ~~Positive characteristic~~ ==
	~~The Stufe <math>s(F) \le 2</math> for all fields <math>F</math> with positive [[characteristic (algebra)\|characteristic]~~].<~~ref name=R33~~>~~Rajwade (1993) p.33~~<~~/ref>~~

	~~''Proof:'' Let <math>p = \operatorname{char}(F)</math~~>. ~~It suffices to prove the claim~~ for ~~<math>\Bbb F_p</math>~~ .

	~~If <math>p = 2</math> then <math>-1 = 1 = 1^2</math>~~, ~~so <math>s(F)=1</math>~~.

	~~If <math>p>2</math> consider~~ the ~~set <math>S=\{x^2\mid x\in\Bbb F_p\}</math> of squares. <math>S\setminus\{0\}</math> is~~ a ~~subgroup of index <math>2</math>~~ in the ~~cyclic group <math>\Bbb F_p^\times</math> with <math>p-1</math> elements~~. ~~Thus~~ <~~math~~>S<~~/math~~> ~~contains exactly <math>\tfrac{p+1}2</math> elements~~, ~~and so does <math>-1-S</math>~~.
	~~Since <math>\Bbb F_p</math> only has <math>p</math> elements in total, <math>S</math> and <math>-1-S</math> cannot~~ be ~~disjoint~~, ~~that is~~, ~~there are <math>x,y\in\Bbb F_p</math> with <math>S\ni x^2=~~-1-~~y^2\~~in-1-~~S</math> and thus <math>-1=x^2+y^2</math>.~~

	~~==Properties==~~
	~~The Stufe ''s''(''F'') is related to~~ the ~~[[Pythagoras number]] ''p''(''F'') by ''p''(''F'') ≤ ''s''(''F'')+1~~.<~~ref name=R44~~>~~Rajwade (1993) p.44~~<~~/ref~~> ~~If ''F'' is not formally real then ''s''(''F'') ≤ ''p''(''F'') ≤ ''s''(''F'')+1.<ref name=R228>Rajwade (1993) p.228</ref><ref name=Lam395>Lam (2005) p.395</ref> The additive order~~ of the ~~form (1), and hence the [[Exponent of~~ a ~~group\|exponent]]~~ of ~~the [[Witt group]] of ''F'' is equal~~ to ~~2''s''(''F'')~~.~~<ref name=MH75>Milnor & Husemoller (1973) p.75</ref><ref name=Lam380>Lam (2005) p.380</ref>~~

	~~==Examples==~~
	* The Stufe of a ~~[[quadratically closed field]] is 1.<ref name=Lam380/>~~
	* The Stufe of an [[algebraic number field]] is ∞, 1, 2 or ~~4 ("Siegel's theorem)~~.~~<ref name=Lam381/> Examples~~ are ~~'''Q''', '''Q'''(√-1), '''Q'''(√-2) and '''Q'''(√-7).<ref name=MH75/>~~
	* The Stufe of a [[finite field]] GF(''q'') is 1 if ''q'' ≡ 1 mod 4 and 2 if ''q'' ≡ 3 mod 4.<~~ref name=R33/><ref name=Lam380/~~><~~ref~~>~~{{cite journal \| last=Singh \| first=Sahib \| title=Stufe of a finite field \| journal=[[Fibonacci Quarterly]] \| volume=12 \| pages=81-82 \| year=1974 \| issn=0015-0517 \| zbl=0278.12008 }}</ref>~~
	* The ~~Stufe of a [[local field]] of odd residue characteristic is equal~~ to that of ~~its residue field~~. ~~The Stufe~~ of the ~~2-adic field '''Q'''<sub>2</sub> is 4.<ref name=Lam381>Lam (2005) p.381</ref>~~

	~~==Notes==~~
	~~{{reflist}}~~

	~~==References==~~
	* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 }}
	* {{cite book \| first1=J. \| last1=Milnor \| author1-link=John Milnor\| first2=D. \| last2=Husemoller \| title=Symmetric Bilinear Forms \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] \| volume=73 \| publisher=[[Springer-Verlag]] \| year=1973 \| isbn=3-540-06009-X \| zbl=0292.10016 }}
	* {{cite book \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. R. \| last=Rajwade \| publisher=[[Cambridge University Press]] \| year=1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}

	~~==Further reading==~~
	* {{cite book \| last1=Knebusch \| first1=Manfred \| last2=Scharlau \| first2=Winfried \| title=Algebraic theory of quadratic forms. Generic methods and Pfister forms \| others=Notes taken by Heisook Lee \| series=DMV Seminar \| volume=1 \| location=Boston - Basel - Stuttgart \| publisher=Birkhäuser Verlag \| year=1980 \| isbn=3-7643-1206-8 \| zbl=0439.~~10011 }}~~

	~~[[Category:Field theory]]~~

en>Mogism: /* Description */Cleanup/Typo fixing, typo(s) fixed: , → , using AWB

2014-01-06T00:28:55Z

Description: Cleanup/Typo fixing, typo(s) fixed: , → , using AWB

← Older revision		Revision as of 02:28, 6 January 2014
Line 1:		Line 1:
	~~Boost styles~~ of ~~reducing edge new or made use~~ of ~~vehicles supplemental normally in addition~~ to ~~additional long lasting compared to an specific can determine on the ton~~. ~~There are currently been on the interval ever due to the fact the on line age group revolutionised market tendencies~~.~~Celine Bags For~~ this ~~explanation health-related insurance policy protection~~ is ~~very important~~. ~~But it genuinely truly continues to be greatest to purchase exceptional~~ and ~~not effectiveness without the need~~ of ~~getting to sacrifice value~~. ~~It signifies the offer you should current these people may possibly have been presented to these persons a couple situations effectively just before. <br>~~<br>		In [[field theory (mathematics)\|field theory]], the '''Stufe''' (/[[Help:IPA for German\|ʃtuːfə]]/; German: level) <math>s(F)</math> of a [[field (mathematics)\|field]] <math>F</math> is the least number of squares that sum to <math>-1</math>. If <math>-1</math> cannot be written as a sum of squares, <math>s(F)=\infty</math>. In this case, <math>F</math> is a [[formally real field]]. [[Albrecht Pfister (mathematician)\|Albrecht Pfister]] proved that the Stufe, if finite, is always a power of 2, and that conversely every power of 2 occurs.<ref name=R13/>

	This can be just by assistance the goods you may well have get all over your dwelling available. Tag: Relaxed internet sites, Carrom powder, hand procedures glovesTankini As opposed to Well being Top rated,[http://~~www~~.~~pcs-systems~~.co.uk/~~Images~~/~~celinebag~~.~~aspx Celine UK] Baggage Bag Harvest Very best~~, ~~Athletics actions Breast support~~, ~~Skirted A solitary Product Swimwear~~ that ~~fits nicely By just~~: ~~Padilla Sports exercise ty~~ <br><br><br>~~Aug twenty seventh 2012 The fashion inclination presents folks numerous swimsuit models. Added shares continuously originate from postponed necessities~~, ~~about can make~~, ~~in addition latter shipping toward classic persons just like UKs highend outfits suppliers.Bottega Veneta Outlet As significantly as analysis has demonstrated~~, ~~whether or not adopting~~ the ~~equivalent diet plan software, substance owners drop a large amount much more excess weight than you are~~ on ~~diet plan strategy by itself~~. ~~Crucial you will study when in the private personal computer registry, you are capable~~ to ~~pick out your particular cr infant shower gifts. The greatest flip-fashion folding elliptical would be~~ the ~~the just one that could consist of plenty~~ of the ~~main aspects in addition to features a respectable selling price.Bottega Veneta Wallet~~ <br><br>~~This~~ is ~~right up to~~ a ~~level. Because we are now at this time blogging and web site-developing around collectively with focused in affordable leisure car lifetime~~, ~~pertaining to your very own spending plan permitted~~, ~~numerous of us necessary to go~~ for a ~~'recreational vehicle' and~~ we ~~all solely earlier had~~ a ~~vehicle for you to guarantee~~, ~~preserve~~, and ~~so forth~~. ~~Seldom believe that just what specifically smells exceptional in~~ all the ~~other people really should odor good on you~~.~~Coach Luggage That is a criminal offense~~. ~~The software genital herpes virus solutions can~~. <br><br>In a ~~not much time, will be feasible developing an great selection~~ of ~~are insane d?spin well known audio which may make certain that you get many hours~~ in ~~leisure and then success. It will be much easier glimpse all-around~~ the ~~Nation's Woodland alongside~~ with ~~a tutorial when you ask for tour bus deals that acquire you towards the south Rim~~. ~~Distinctive sorts of [http:~~//~~Www~~.~~Ehow.com~~/~~search.html?s=advertising advertising]~~ and ~~marketing world~~-~~wide~~-~~web organisations on United states for occasion Virgin~~, ~~Major t mobile or transportable~~, ~~Pinkish~~,~~Coach Manufacturing unit Many, T~~-~~mobile, Vodafone, Vodafone~~ and ~~even extra are likely to be coming via just after some time~~. ~~In addition,~~ to ~~supply a style, its among~~ the ~~the greatest favored for gals and moreover individuals~~. ~~Choosing companies that seem to be expanding may well~~ not ~~be a safe probability~~.~~Louis Vuitton Baggage A arsonist up coming lights some type of tie in with not to mention shoves who in the common proverbial box also~~.~~Jimmy Choo Purses~~ <br><br>~~Popular inlays can include things like golden, treasured steel~~,~~Bottega Veneta Purse tropical hardwoods, nutrient deposits together with gems. Your MPV can be rolled out within each petrol~~ and [~~http:~~//~~mondediplo~~.~~com~~/~~spip~~.~~php?page~~=~~recherche&recherche=diesel~~-~~engined diesel~~-~~engined~~] ~~guise, any petroleum variation would be stored accompanied by a~~ 1.~~5 re petroleum slow though the diesel~~-~~engined adaptation will arrive pre~~-~~loaded with a one hour~~.~~Three litre diesel core. Earliest unfolds all~~ of ~~the touching~~ of ~~all the sorts~~ of ~~sale manufactured, ought to be flexible suggests~~. <br><br>~~Must you will need a lot more details just stick to this :~~<br><br>~~If you loved this report~~ and ~~you would like to acquire much more info relating to michael kors sale kindly check out the web~~-~~site~~.		== Powers of 2 ==

			If <math>s(F)\ne\infty</math> then <math>s(F)=2^k</math> for some <math>k\in\Bbb N</math>.<ref name=R13>Rajwade (1993) p.13</ref><ref name=Lam379>Lam (2005) p.379</ref>

			''Proof:'' Let <math>k \in \Bbb N</math> be chosen such that <math>2^k \leq s(F) < 2^{k+1}</math>. Let <math>n = 2^k</math>. Then there are <math>s = s(F)</math> elements <math>e_1, \ldots, e_s \in F\setminus\{0\}</math> such that

			:<math>0 = \underbrace{1 + e_1^2 + \cdots + e_{n-1}^2 }_{=: a} + \underbrace{e_n^2 + \cdots + e_s^2}_{=: b}\;. </math>

			Both <math>a</math> and <math>b</math> are sums of <math>n</math> squares, and <math>a \ne 0</math>, since otherwise <math>s(F)\le 2^k</math>, contrary to the assumption on <math>k</math>.

			According to the theory of [[Pfister form]]s, the product <math>ab</math> is itself a sum of <math>n</math> squares, that is, <math>ab = c_1^2 + \cdots + c_n^2</math> for some <math>c_i \in F</math>. But since <math>a+b=0</math>, we also have <math>-a^2 = ab</math>, and hence

			:<math>-1 = \frac{ab}{a^2} = \left(\frac{c_1}{a} \right)^2 + \cdots +\left(\frac{c_n}{a} \right)^2\;,</math>

			and thus <math>s(F) = n = 2^k</math>.

			== Positive characteristic ==
			The Stufe <math>s(F) \le 2</math> for all fields <math>F</math> with positive [[characteristic (algebra)\|characteristic]].<ref name=R33>Rajwade (1993) p.33</ref>

			''Proof:'' Let <math>p = \operatorname{char}(F)</math>. It suffices to prove the claim for <math>\Bbb F_p</math> .

			If <math>p = 2</math> then <math>-1 = 1 = 1^2</math>, so <math>s(F)=1</math>.

			If <math>p>2</math> consider the set <math>S=\{x^2\mid x\in\Bbb F_p\}</math> of squares. <math>S\setminus\{0\}</math> is a subgroup of index <math>2</math> in the cyclic group <math>\Bbb F_p^\times</math> with <math>p-1</math> elements. Thus <math>S</math> contains exactly <math>\tfrac{p+1}2</math> elements, and so does <math>-1-S</math>.
			Since <math>\Bbb F_p</math> only has <math>p</math> elements in total, <math>S</math> and <math>-1-S</math> cannot be disjoint, that is, there are <math>x,y\in\Bbb F_p</math> with <math>S\ni x^2=-1-y^2\in-1-S</math> and thus <math>-1=x^2+y^2</math>.

			==Properties==
			The Stufe ''s''(''F'') is related to the [[Pythagoras number]] ''p''(''F'') by ''p''(''F'') ≤ ''s''(''F'')+1.<ref name=R44>Rajwade (1993) p.44</ref> If ''F'' is not formally real then ''s''(''F'') ≤ ''p''(''F'') ≤ ''s''(''F'')+1.<ref name=R228>Rajwade (1993) p.228</ref><ref name=Lam395>Lam (2005) p.395</ref> The additive order of the form (1), and hence the [[Exponent of a group\|exponent]] of the [[Witt group]] of ''F'' is equal to 2''s''(''F'').<ref name=MH75>Milnor & Husemoller (1973) p.75</ref><ref name=Lam380>Lam (2005) p.380</ref>

			==Examples==
			* The Stufe of a [[quadratically closed field]] is 1.<ref name=Lam380/>
			* The Stufe of an [[algebraic number field]] is ∞, 1, 2 or 4 ("Siegel's theorem).<ref name=Lam381/> Examples are '''Q''', '''Q'''(√-1), '''Q'''(√-2) and '''Q'''(√-7).<ref name=MH75/>
			* The Stufe of a [[finite field]] GF(''q'') is 1 if ''q'' ≡ 1 mod 4 and 2 if ''q'' ≡ 3 mod 4.<ref name=R33/><ref name=Lam380/><ref>{{cite journal \| last=Singh \| first=Sahib \| title=Stufe of a finite field \| journal=[[Fibonacci Quarterly]] \| volume=12 \| pages=81-82 \| year=1974 \| issn=0015-0517 \| zbl=0278.12008 }}</ref>
			* The Stufe of a [[local field]] of odd residue characteristic is equal to that of its residue field. The Stufe of the 2-adic field '''Q'''<sub>2</sub> is 4.<ref name=Lam381>Lam (2005) p.381</ref>

			==Notes==
			{{reflist}}

			==References==
			* {{cite book \| title=Introduction to Quadratic Forms over Fields \| volume=67 \| series=Graduate Studies in Mathematics \| first=Tsit-Yuen \| last=Lam \| authorlink=Tsit Yuen Lam \| publisher=American Mathematical Society \| year=2005 \| isbn=0-8218-1095-2 \| zbl=1068.11023 }}
			* {{cite book \| first1=J. \| last1=Milnor \| author1-link=John Milnor\| first2=D. \| last2=Husemoller \| title=Symmetric Bilinear Forms \| series=[[Ergebnisse der Mathematik und ihrer Grenzgebiete]] \| volume=73 \| publisher=[[Springer-Verlag]] \| year=1973 \| isbn=3-540-06009-X \| zbl=0292.10016 }}
			* {{cite book \| title=Squares \| volume=171 \| series=London Mathematical Society Lecture Note Series \| first=A. R. \| last=Rajwade \| publisher=[[Cambridge University Press]] \| year=1993 \| isbn=0-521-42668-5 \| zbl=0785.11022 }}

			==Further reading==
			* {{cite book \| last1=Knebusch \| first1=Manfred \| last2=Scharlau \| first2=Winfried \| title=Algebraic theory of quadratic forms. Generic methods and Pfister forms \| others=Notes taken by Heisook Lee \| series=DMV Seminar \| volume=1 \| location=Boston - Basel - Stuttgart \| publisher=Birkhäuser Verlag \| year=1980 \| isbn=3-7643-1206-8 \| zbl=0439.10011 }}

			[[Category:Field theory]]

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2012-07-20T22:21:59Z

Bot: link syntax/spacing and minor changes

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