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In [[mathematics]], a '''Pfister form''' is a particular kind of [[quadratic form]] over a [[field (mathematics)|field]] ''F'' (whose [[characteristic (algebra)|characteristic]] is usually assumed to be not 2), introduced by [[Albrecht Pfister (mathematician)|Albrecht Pfister]] in 1965. A Pfister form is in 2<sup>''n''</sup> variables, for some natural number ''n'' (also called an '''n-Pfister form'''), and may be written as a [[tensor product of quadratic forms]] as: | |||
:<math>\langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle \cong \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle,</math> | |||
for ''a<sub>i</sub>'' elements of the field ''F''. An ''n''-Pfister form may also be constructed inductively from an (''n''-1)-Pfister form ''q'' and an element ''a'' of ''F'', as <math>q \oplus (a)q</math>. | |||
So all 1-Pfister forms and 2-Pfister forms look like: | |||
:<math>\langle\!\langle a\rangle\!\rangle\cong \langle 1, a \rangle \cong x^2 + ay^2</math>. | |||
:<math>\langle\!\langle a,b\rangle\!\rangle\cong \langle 1, a, b, ab \rangle \cong x^2 + ay^2 +bz^2 +abw^2.</math> | |||
For n ≤ 3 the ''n''-Pfister forms are [[norm form]]s of [[composition algebra]]s.<ref name=Lam316>Lam (2005) p.316</ref> In fact, in this case, two ''n''-Pfister forms are [[Isometry|isometric]] if and only if the corresponding composition algebras are [[isomorphic]]. | |||
The Pfister forms are generators for the torsion in the [[Witt group]].<ref name=Lam395>Lam (2005) p.395</ref> The ''n''-fold forms additively generate the ''n''-th power ''I''<sup>''n''</sup> of the fundamental ideal of the Witt ring.<ref name=Lam316>Lam (2005) p.316</ref> | |||
==Characterisation== | |||
We define a quadratic form ''q'' over a field ''F'' to be '''multiplicative''' if when '''x''' and '''y''' are vectors of indeterminates, then ''q''('''x''').''q''('''y''') = ''q''('''z''') where '''z''' is a vector of [[rational function]]s in the '''x''' and '''y''' over ''F''. [[Isotropic quadratic form]]s are multiplicative.<ref name=Lam324>Lam (2005) p.324</ref> For [[anisotropic quadratic form]]s, Pfister forms are multiplicative and conversely.<ref name=Lam325>Lam (2005) p.325</ref><ref name=Raj164>Rajwade (1993) p.164</ref> | |||
==Connection with K-theory== | |||
Let ''k''<sub>''n''</sub>(''F'') be the ''n''-th group in [[Milnor K-theory]] modulo 2. There are homomorphisms from ''k''<sub>''n''</sub>(''F'') to the Witt ring by taking the symbol | |||
:<math> \{a_1,\ldots,a_n\} \mapsto \langle\!\langle a_1, a_2, ... , a_n \rangle\!\rangle ,</math> | |||
where the image is an ''n''-fold Pfister form</sup>.<ref name=Lam366>Lam (2005) p.366</ref> The image can be taken as ''I''<sup>''n''</sup>/''I''<sup>''n''+1</sup> and the map is surjective since the Pfister forms additively generate ''I''<sup>''n''</sup>. The [[Milnor conjecture]] can be interpreted as stating that these maps are isomorphisms.<ref name=Lam366/> | |||
==Pfister neighbours== | |||
A '''Pfister neighbour''' is a form (''W'',σ) such that (''W'',σ) is similar to a subspace of a space with Pfister form (''V'',φ) where dim.''V'' < 2 dim.''W''.<ref name=Lam339>Lam (2005) p.339</ref> The associated Pfister form φ is uniquely determined by σ. Any ternary form is a Pfister neighbour; a quaternary form is a Pfister neighbour if and only if its discriminant is a square.<ref name=Lam341>Lam (2005) p.341</ref> A degree five form is a Pfister neighbour if and only if the underlying field is a [[linked field]].<ref name=Lam342>Lam (2005) p.342</ref> | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
* {{Citation | 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 | mr = 2104929 }}, Ch. 10 | |||
* {{citation | 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== | |||
* {{citation | 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:Quadratic forms]] | |||
Revision as of 04:31, 21 November 2013
In mathematics, a Pfister form is a particular kind of quadratic form over a field F (whose characteristic is usually assumed to be not 2), introduced by Albrecht Pfister in 1965. A Pfister form is in 2n variables, for some natural number n (also called an n-Pfister form), and may be written as a tensor product of quadratic forms as:
for ai elements of the field F. An n-Pfister form may also be constructed inductively from an (n-1)-Pfister form q and an element a of F, as .
So all 1-Pfister forms and 2-Pfister forms look like:
For n ≤ 3 the n-Pfister forms are norm forms of composition algebras.[1] In fact, in this case, two n-Pfister forms are isometric if and only if the corresponding composition algebras are isomorphic.
The Pfister forms are generators for the torsion in the Witt group.[2] The n-fold forms additively generate the n-th power In of the fundamental ideal of the Witt ring.[1]
Characterisation
We define a quadratic form q over a field F to be multiplicative if when x and y are vectors of indeterminates, then q(x).q(y) = q(z) where z is a vector of rational functions in the x and y over F. Isotropic quadratic forms are multiplicative.[3] For anisotropic quadratic forms, Pfister forms are multiplicative and conversely.[4][5]
Connection with K-theory
Let kn(F) be the n-th group in Milnor K-theory modulo 2. There are homomorphisms from kn(F) to the Witt ring by taking the symbol
where the image is an n-fold Pfister form.[6] The image can be taken as In/In+1 and the map is surjective since the Pfister forms additively generate In. The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.[6]
Pfister neighbours
A Pfister neighbour is a form (W,σ) such that (W,σ) is similar to a subspace of a space with Pfister form (V,φ) where dim.V < 2 dim.W.[7] The associated Pfister form φ is uniquely determined by σ. Any ternary form is a Pfister neighbour; a quaternary form is a Pfister neighbour if and only if its discriminant is a square.[8] A degree five form is a Pfister neighbour if and only if the underlying field is a linked field.[9]
Notes
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References
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010, Ch. 10 - Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
Further reading
- Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010