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In [[mathematics]], a [[field (mathematics)|field]] ''F'' is called '''quasi-algebraically closed''' (or '''C<sub>1</sub>''') if every non-constant [[homogeneous polynomial]] ''P'' over ''F'' has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by [[C. C. Tsen]], a student of [[Emmy Noether]] in a 1936 paper; and later in the 1951 [[Princeton University]] dissertation of [[Serge Lang]]. The idea itself is attributed to Lang's advisor [[Emil Artin]].
 
Formally, if ''P'' is a non-constant homogeneous polynomial in variables
 
:''X''<sub>1</sub>, ..., ''X''<sub>''N''</sub>,
 
and of degree ''d'' satisfying
 
:''d'' < ''N''
 
then it has a non-trivial zero over ''F''; that is, for some ''x''<sub>''i''</sub> in ''F'', not all 0, we have
 
:''P''(''x''<sub>''1''</sub>, ..., ''x''<sub>''N''</sub>) = 0.
 
In geometric language, the [[hypersurface]] defined by ''P'', in [[projective space]] of dimension ''N'' &minus; 2,  then has a point over ''F''.
 
==Examples==
*Any [[algebraically closed field]] is quasi-algebraically closed. In fact, any homogeneous polynomial in at least two variables over an algebraically closed field has a non-trivial zero.<ref name=FJ455>Fried & Jarden (2008) p.455</ref>
*Any [[finite field]] is quasi-algebraically closed by the [[Chevalley–Warning theorem]].<ref name=FJ456>Fried & Jarden (2008) p.456</ref><ref name=S79162>Serre (1979) p.162</ref><ref name=GS142>Gille & Szamuley (2006) p.142</ref>
*[[Algebraic function field]]s over algebraically closed fields are quasi-algebraically closed by [[Tsen's theorem]].<ref name=S79162/><ref name=GS143>Gille & Szamuley (2006) p.143</ref>
*The maximal unramified extension of a complete field with a discrete valuation and a [[perfect field|perfect]] residue field is quasi-algebraically closed.<ref name=S79162/>
*A complete field with a discrete valuation and an algebraically closed residue field is quasi-algebraically closed by a result of Lang.<ref name=S79162/><ref name=GS144>Gille & Szamuley (2006) p.144</ref>
* A [[pseudo algebraically closed field]] of [[Characteristic (algebra)|characteristic]] zero is quasi-algebraically closed.<ref name=FJ462>Fried & Jarden (2008) p.462</ref>
 
==Properties==
*Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.
*The [[Brauer group]] of a finite extension of a quasi-algebraically closed field is trivial.<ref>Lorenz (2008) p.181</ref><ref name=S79161>Serre (1979) p.161</ref><ref name=GS141>Gille & Szamuely (2006) p.141</ref>
*A quasi-algebraically closed field has [[Cohomological dimension of a field|cohomological dimension]] at most 1.<ref name=GS141/>
 
==''C''<sub>k</sub> fields==
Quasi-algebraically closed fields are also called ''C''<sub>1</sub>. A '''C<sub>''k''</sub> field''', more generally, is one for which any homogeneous polynomial of degree ''d'' in ''N'' variables has a non-trivial zero, provided
 
:''d''<sup>''k''</sup> < ''N'',
 
for ''k'' &ge; 1.<ref name=SGC87>Serre (1997) p.87</ref>  The condition was first introduced and studied by Lang.<ref name=GS141/>  If a field is C<sub>i</sub> then so is a finite extension.<ref name=L245>Lang (1997) p.245</ref><ref name=SGC87/>  The C<sub>0</sub> fields are precisely the algebraically closed fields.<ref name=NSW361/><ref name=Lor116>Lorenz (2008) p.116</ref>
 
Lang and Nagata proved that if a field is ''C''<sub>''k''</sub>, then any extension of [[transcendence degree]] ''n'' is ''C''<sub>''k''+''n''</sub>.<ref name=Lor119>Lorenz (2008) p.119</ref><ref name=SGC88>Serre (1997) p.88</ref><ref name=FJ459>Fried & Jarden (2008) p.459</ref>  The smallest ''k'' such that ''K'' is a ''C''<sub>k</sub> field (<math>\infty</math> if no such number exists), is called the '''[[diophantine dimension]]''' ''dd''(''K'') of ''K''.<ref name=NSW361>{{cite book | title=Cohomology of Number Fields | volume=323 | series=Grundlehren der Mathematischen Wissenschaften | first1=Jürgen | last1=Neukirch | first2=Alexander | last2=Schmidt | first3=Kay | last3=Wingberg | edition=2nd | publisher=[[Springer-Verlag]] | year=2008 | isbn=3-540-37888-X | page=361}}</ref>
 
===''C''<sub>2</sub> fields===
 
Every finite field is C<sub>2</sub>.<ref name=FJ462/>
 
====Properties====
Suppose that the field ''k'' is ''C''<sub>2</sub>.
* Any skew field ''D'' finite over ''k'' as centre has the property that the [[reduced norm]] ''D''<sup>&lowast;</sup> → ''k''<sup>&lowast;</sup> is surjective.<ref name=SGC88/>
* Every quadratic form in 5 or more variables over ''k'' is [[Isotropic quadratic form|isotropic]].<ref name=SGC88/>
 
====Artin's conjecture====
Artin conjectured that [[p-adic field|''p''-adic field]]s were ''C''<sub>2</sub>, but
[[Guy Terjanian]] found [[Ax–Kochen theorem|''p''-adic counterexamples]] for all ''p''.<ref>{{cite journal | first=Guy | last=Terjanian | authorlink=Guy Terjanian | title=Un contre-example à une conjecture d'Artin | journal=C. R. Acad. Sci. Paris Sér. A-B | volume=262 | page=A612 | year=1966 | zbl=0133.29705 | language=French }}</ref><ref name=L247>Lang (1997) p.247</ref>  The [[Ax–Kochen theorem]]  applied methods from [[model theory]] to show that Artin's conjecture was true for '''Q'''<sub>''p''</sub> with ''p'' large enough (depending on ''d'').
 
===Weakly C<sub>''k''</sub> fields===
A field ''K'' is '''weakly C<sub>''k'',''d''</sub>''' if for every homogeneous polynomial of degree ''d'' in ''N'' variables satisfying
:''d''<sup>''k''</sup> < ''N''
the [[Zariski topology|Zariski closed]] set ''V''(''f'') of '''P'''<sup>''n''</sub>(''K'') contains a [[subvariety]] which is Zariski closed over ''K''.
 
A field which is weakly C<sub>''k'',''d''</sub> for every ''d'' is '''weakly C<sub>''k''</sub>'''.<ref name=FJ456/> 
 
====Properties====
* A C<sub>''k''</sub> field is weakly C<sub>''k''</sub>.<ref name=FJ456/>
* A [[perfect field|perfect]] PAC weakly C<sub>''k''</sub> field is C<sub>''k''</sub>.<ref name=FJ456/>
* A field ''K'' is weakly C<sub>''k'',''d''</sub> if and only if every form satisfying the conditions has a point '''x''' defined over a field which is a [[primary extension]] of ''K''.<ref name=FJ457>Fried & Jarden (2008) p.457</ref>
* If a field is weakly C<sub>''k''</sub>, then any extension of transcendence degree ''n'' is weakly C<sub>''k''+''n''</sub>.<ref name=FJ459/>
 
* Any extension of an algebraically closed field is weakly C<sub>1</sub>.<ref name=FJ461/>
* Any field with procyclic absolute Galois group is weakly C<sub>1</sub>.<ref name=FJ461/>
* Any field of positive characteristic is weakly C<sub>2</sub>.<ref name=FJ461/>
 
* If the field of rational numbers is weakly C<sub>1</sub>, then every field is weakly C<sub>1</sub>.<ref name=FJ461>Fried & Jarden (2008) p.461</ref>
 
==See also==
* [[Brauer's theorem on forms]]
* [[Tsen rank]]
 
==References==
{{reflist|2}}
*{{cite journal | first1=James | last1=Ax | author1-link=James Ax | first2=Simon | last2=Kochen | author2-link=Simon B. Kochen | title=Diophantine problems over local fields I | title=Amer. J. Math. | volume=87 | pages=605–630 | year=1965 | zbl=0136.32805 }}
* {{cite book | last1=Fried | first1=Michael D. | last2=Jarden | first2=Moshe | title=Field arithmetic | edition=3rd revised | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge | volume=11 | publisher=[[Springer-Verlag]] | year=2008 | isbn=978-3-540-77269-9 | zbl=1145.12001 }}
* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
* {{cite book | zbl=0185.08304 | last=Greenberg | first=M.J. | title=Lectures of forms in many variables | series=Mathematics Lecture Note Series | location=New York-Amsterdam | publisher=W.A. Benjamin | year=1969 | zbl=0185.08304| series=Mathematics Lecture Note Series }}
* {{cite journal | zbl=0046.26202 | last=Lang | first=Serge | authorlink=Serge Lang | title=On quasi algebraic closure | journal=[[Annals of Mathematics]] | volume=55 | year=1952 | pages=373–390 }}
* {{cite book | first=Serge | last=Lang | authorlink=Serge Lang | title=Survey of Diophantine Geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
* {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | pages=109-126 | zbl=1130.12001 }}
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Local fields | others=Translated from the French by Marvin Jay Greenberg | series=[[Graduate Texts in Mathematics]] | volume=67 | publisher=[[Springer-Verlag]] | year=1979 | isbn=0-387-90424-7 | zbl=0423.12016 }}
* {{cite book | last=Serre | first=Jean-Pierre | authorlink=Jean-Pierre Serre | title=Galois cohomology | publisher=[[Springer-Verlag]] | year=1997| isbn=3-540-61990-9 | zbl=0902.12004 }}
*{{cite journal | first=C. | last=Tsen | authorlink=C. C. Tsen | title=Zur Stufentheorie der Quasi-algebraisch-Abgeschlossenheit kommutativer Körper | journal=J. Chinese Math. Soc. | volume=171 | year=1936 | pages=81–92 | zbl=0015.38803 }}
 
[[Category:Field theory]]
[[Category:Diophantine geometry]]

Revision as of 01:06, 22 December 2013

In mathematics, a field F is called quasi-algebraically closed (or C1) if every non-constant homogeneous polynomial P over F has a non-trivial zero provided the number of its variables is more than its degree. The idea of quasi-algebraically closed fields was investigated by C. C. Tsen, a student of Emmy Noether in a 1936 paper; and later in the 1951 Princeton University dissertation of Serge Lang. The idea itself is attributed to Lang's advisor Emil Artin.

Formally, if P is a non-constant homogeneous polynomial in variables

X1, ..., XN,

and of degree d satisfying

d < N

then it has a non-trivial zero over F; that is, for some xi in F, not all 0, we have

P(x1, ..., xN) = 0.

In geometric language, the hypersurface defined by P, in projective space of dimension N − 2, then has a point over F.

Examples

Properties

  • Any algebraic extension of a quasi-algebraically closed field is quasi-algebraically closed.
  • The Brauer group of a finite extension of a quasi-algebraically closed field is trivial.[8][9][10]
  • A quasi-algebraically closed field has cohomological dimension at most 1.[10]

Ck fields

Quasi-algebraically closed fields are also called C1. A Ck field, more generally, is one for which any homogeneous polynomial of degree d in N variables has a non-trivial zero, provided

dk < N,

for k ≥ 1.[11] The condition was first introduced and studied by Lang.[10] If a field is Ci then so is a finite extension.[12][11] The C0 fields are precisely the algebraically closed fields.[13][14]

Lang and Nagata proved that if a field is Ck, then any extension of transcendence degree n is Ck+n.[15][16][17] The smallest k such that K is a Ck field (∞ if no such number exists), is called the diophantine dimension dd(K) of K.[13]

C2 fields

Every finite field is C2.[7]

Properties

Suppose that the field k is C2.

  • Any skew field D finite over k as centre has the property that the reduced norm D∗ → k∗ is surjective.[16]
  • Every quadratic form in 5 or more variables over k is isotropic.[16]

Artin's conjecture

Artin conjectured that p-adic fields were C2, but Guy Terjanian found p-adic counterexamples for all p.[18][19] The Ax–Kochen theorem applied methods from model theory to show that Artin's conjecture was true for Qp with p large enough (depending on d).

Weakly Ck fields

A field K is weakly Ck,d if for every homogeneous polynomial of degree d in N variables satisfying

dk < N

the Zariski closed set V(f) of Pn(K) contains a subvariety which is Zariski closed over K.

A field which is weakly Ck,d for every d is weakly Ck.[2]

Properties

  • A Ck field is weakly Ck.[2]
  • A perfect PAC weakly Ck field is Ck.[2]
  • A field K is weakly Ck,d if and only if every form satisfying the conditions has a point x defined over a field which is a primary extension of K.[20]
  • If a field is weakly Ck, then any extension of transcendence degree n is weakly Ck+n.[17]
  • Any extension of an algebraically closed field is weakly C1.[21]
  • Any field with procyclic absolute Galois group is weakly C1.[21]
  • Any field of positive characteristic is weakly C2.[21]
  • If the field of rational numbers is weakly C1, then every field is weakly C1.[21]

See also

References

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  1. ↑ Fried & Jarden (2008) p.455
  2. ↑ 2.0 2.1 2.2 2.3 Fried & Jarden (2008) p.456
  3. ↑ 3.0 3.1 3.2 3.3 Serre (1979) p.162
  4. ↑ Gille & Szamuley (2006) p.142
  5. ↑ Gille & Szamuley (2006) p.143
  6. ↑ Gille & Szamuley (2006) p.144
  7. ↑ 7.0 7.1 Fried & Jarden (2008) p.462
  8. ↑ Lorenz (2008) p.181
  9. ↑ Serre (1979) p.161
  10. ↑ 10.0 10.1 10.2 Gille & Szamuely (2006) p.141
  11. ↑ 11.0 11.1 Serre (1997) p.87
  12. ↑ Lang (1997) p.245
  13. ↑ 13.0 13.1 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

    My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
  14. ↑ Lorenz (2008) p.116
  15. ↑ Lorenz (2008) p.119
  16. ↑ 16.0 16.1 16.2 Serre (1997) p.88
  17. ↑ 17.0 17.1 Fried & Jarden (2008) p.459
  18. ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

    In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

    Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

    Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

    A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

    The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

    There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
  19. ↑ Lang (1997) p.247
  20. ↑ Fried & Jarden (2008) p.457
  21. ↑ 21.0 21.1 21.2 21.3 Fried & Jarden (2008) p.461
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