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In [[algebra]], a [[field (mathematics)|field]] ''k'' is said to be '''perfect''' if any one of the following equivalent conditions holds: | |||
* Every [[irreducible polynomial]] over ''k'' has distinct roots. | |||
* Every [[irreducible polynomial]] over ''k'' is [[separable polynomial|separable]]. | |||
* Every [[finite extension]] of ''k'' is [[separable extension|separable]]. | |||
* Every [[algebraic extension]] of ''k'' is separable. | |||
* Either ''k'' has [[characteristic (algebra)|characteristic]] 0, or, when ''k'' has characteristic ''p'' > 0, every element of ''k'' is a [[Power (mathematics)|''p''th power]]. | |||
* Either ''k'' has [[characteristic (algebra)|characteristic]] 0, or, when ''k'' has characteristic ''p'' > 0, the [[Frobenius endomorphism]] ''x''→''x''<sup>''p''</sup> is an [[automorphism]] of ''k'' | |||
* The [[separable closure]] of ''k'' is [[algebraically closed]]. | |||
* Every [[reduced ring|reduced]] commutative [[Algebra (ring theory)|''k''-algebra]] ''A'' is a [[separable algebra]]; i.e., <math>A \otimes_k F</math> is [[reduced ring|reduced]] for every [[field extension]] ''F''/''k''. (see below) | |||
Otherwise, ''k'' is called '''imperfect'''. | |||
In particular, all fields of characteristic zero and all [[finite field]]s are perfect. | |||
Perfect fields are significant because [[Galois theory]] over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above). | |||
More generally, a [[ring (mathematics)|ring]] of characteristic ''p'' (''p'' a [[prime number|prime]]) is called '''perfect''' if the [[Frobenius endomorphism]] is an [[automorphism]].<ref>{{harvnb|Serre|1979}}, Section II.4</ref> (This is equivalent to the above condition "every element of ''k'' is a ''p''th power" for integral domains.) | |||
==Examples== | |||
Examples of perfect fields are: | |||
* every field of characteristic zero, e.g. the field of [[rational numbers]] or the field of [[complex numbers]]; | |||
* every [[finite field]], e.g. the field '''F'''<sub>''p''</sub> = '''Z'''/''p'''''Z''' where ''p'' is a [[prime number]]; | |||
* every [[algebraically closed field]]; | |||
* the union of a set of perfect fields totally ordered by extension; | |||
* fields algebraic over a perfect field. | |||
In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic ''p''>0. Every imperfect field is necessarily [[Field_extension#Algebraic_and_transcendental_elements|transcendental]] over its [[prime subfield]] (the minimal subfield), because the latter is perfect. An example of an imperfect field is | |||
*the field <math>k(X)</math> of all rational functions in an indeterminate <math>X</math>, where ''k'' has characteristic ''p''>0 (because ''X'' has no ''p''-th root in ''k''(''X'')). | |||
== Field extension over a perfect field == | |||
Any [[finitely generated field extension]] over a perfect field is [[separably generated field extension|separably generated]].<ref>Matsumura, Theorem 26.2</ref> | |||
==Perfect closure and perfection== | |||
One of the equivalent conditions says that, in characteristic ''p'', a field adjoined with all ''p''<sup>''r''</sup>-th roots (''r''≥1) is perfect; it is called the '''perfect closure''' of ''k'' and usually denoted by <math>k^{p^{-\infty}}</math>. | |||
The perfect closure can be used in a test for separability. More precisely, a commutative ''k''-algebra ''A'' is separable if and only if <math>A \otimes_k k^{p^{-\infty}}</math> is reduced.<ref>{{harvnb|Cohn|2003|loc=Theorem 11.6.10}}</ref> | |||
In terms of [[universal property|universal properties]], the '''perfect closure''' of a ring ''A'' of characteristic ''p'' is a perfect ring ''A<sub>p</sub>'' of characteristic ''p'' together with a [[ring homomorphism]] ''u'' : ''A'' → ''A<sub>p</sub>'' such that for any other perfect ring ''B'' of characteristic ''p'' with a homomorphism ''v'' : ''A'' → ''B'' there is a unique homomorphism ''f'' : ''A<sub>p</sub>'' → ''B'' such that ''v'' factors through ''u'' (i.e. ''v'' = ''fu''). The perfect closure always exists; the proof involves "adjoining ''p''-th roots of elements of ''A''", similar to the case of fields.<ref>{{harvnb|Bourbaki|2003}}, Section V.5.1.4, page 111</ref> | |||
The '''perfection''' of a ring ''A'' of characteristic ''p'' is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection ''R''(''A'') of ''A'' is a perfect ring of characteristic ''p'' together with a map θ : ''R''(''A'') → ''A'' such that for any perfect ring ''B'' of characteristic ''p'' equipped with a map φ : ''B'' → ''A'', there is a unique map ''f'' : ''B'' → ''R''(''A'') such that φ factors through θ (i.e. φ = θ''f''). The perfection of ''A'' may be constructed as follows. Consider the [[projective system]] | |||
:<math>\cdots\rightarrow A\rightarrow A\rightarrow A\rightarrow\cdots</math> | |||
where the transition maps are the Frobenius endomorphism. The [[inverse limit]] of this system is ''R''(''A'') and consists of sequences (''x''<sub>0</sub>, ''x''<sub>1</sub>, ... ) of elements of ''A'' such that <math>x_{i+1}^p=x_i</math> for all ''i''. The map θ : ''R''(''A'') → ''A'' sends (''x<sub>i</sub>'') to ''x''<sub>0</sub>.<ref>{{harvnb|Brinon|Conrad|2009}}, section 4.2</ref> | |||
== See also == | |||
*[[p-ring]] | |||
*[[Quasi-finite field]] | |||
==Notes== | |||
{{reflist}} | |||
== References == | |||
*{{Citation | |||
| last=Bourbaki | |||
| first=Nicolas | |||
| author-link=Nicolas Bourbaki | |||
| title=Algebra II | |||
| isbn=978-3-540-00706-7 | |||
| publisher=Springer | |||
| year=2003 | |||
}} | |||
*{{Citation | |||
| last=Brinon | |||
| first=Olivier | |||
| last2=Conrad | |||
| first2=Brian | |||
| author2-link=Brian Conrad | |||
| title=CMI Summer School notes on p-adic Hodge theory | |||
| url=http://math.stanford.edu/~conrad/papers/notes.pdf | |||
| year=2009 | |||
| accessdate=2010-02-05 | |||
}} | |||
*{{Citation | |||
| last=Serre | |||
| first=Jean-Pierre | |||
| author-link=Jean-Pierre Serre | |||
| title=[[Local Fields (book)|Local fields]] | |||
| year=1979 | |||
| edition=2 | |||
| publisher=[[Springer-Verlag]] | |||
| series=[[Graduate Texts in Mathematics]] | |||
| volume=67 | |||
| mr=554237 | |||
| isbn=978-0-387-90424-5 | |||
}} | |||
*{{citation | |||
|last=Cohn | |||
|first =P.M. | |||
|year=2003 | |||
|title=Basic Algebra: Groups, Rings and Fields | |||
}} | |||
*{{citation | |||
|last=Matsumura | |||
|first =H | |||
|year=2003 | |||
|title=Commutative ring theory | |||
|series=Translated from the Japanese by M. Reid. [[Cambridge Studies in Advanced Mathematics]] | |||
|volume=8 | |||
|edition=2nd | |||
}} | |||
==External links== | |||
* {{springer|title=Perfect field|id=p/p072040}} | |||
[[Category:Ring theory]] | |||
[[Category:Field theory]] | |||
Revision as of 16:17, 25 June 2013
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds:
- Every irreducible polynomial over k has distinct roots.
- Every irreducible polynomial over k is separable.
- Every finite extension of k is separable.
- Every algebraic extension of k is separable.
- Either k has characteristic 0, or, when k has characteristic p > 0, every element of k is a pth power.
- Either k has characteristic 0, or, when k has characteristic p > 0, the Frobenius endomorphism x→xp is an automorphism of k
- The separable closure of k is algebraically closed.
- Every reduced commutative k-algebra A is a separable algebra; i.e., is reduced for every field extension F/k. (see below)
Otherwise, k is called imperfect.
In particular, all fields of characteristic zero and all finite fields are perfect.
Perfect fields are significant because Galois theory over these fields becomes simpler, since the general Galois assumption of field extensions being separable is automatically satisfied over these fields (see third condition above).
More generally, a ring of characteristic p (p a prime) is called perfect if the Frobenius endomorphism is an automorphism.[1] (This is equivalent to the above condition "every element of k is a pth power" for integral domains.)
Examples
Examples of perfect fields are:
- every field of characteristic zero, e.g. the field of rational numbers or the field of complex numbers;
- every finite field, e.g. the field Fp = Z/pZ where p is a prime number;
- every algebraically closed field;
- the union of a set of perfect fields totally ordered by extension;
- fields algebraic over a perfect field.
In fact, most fields that appear in practice are perfect. The imperfect case arises mainly in algebraic geometry in characteristic p>0. Every imperfect field is necessarily transcendental over its prime subfield (the minimal subfield), because the latter is perfect. An example of an imperfect field is
- the field of all rational functions in an indeterminate , where k has characteristic p>0 (because X has no p-th root in k(X)).
Field extension over a perfect field
Any finitely generated field extension over a perfect field is separably generated.[2]
Perfect closure and perfection
One of the equivalent conditions says that, in characteristic p, a field adjoined with all pr-th roots (r≥1) is perfect; it is called the perfect closure of k and usually denoted by .
The perfect closure can be used in a test for separability. More precisely, a commutative k-algebra A is separable if and only if is reduced.[3]
In terms of universal properties, the perfect closure of a ring A of characteristic p is a perfect ring Ap of characteristic p together with a ring homomorphism u : A → Ap such that for any other perfect ring B of characteristic p with a homomorphism v : A → B there is a unique homomorphism f : Ap → B such that v factors through u (i.e. v = fu). The perfect closure always exists; the proof involves "adjoining p-th roots of elements of A", similar to the case of fields.[4]
The perfection of a ring A of characteristic p is the dual notion (though this term is sometimes used for the perfect closure). In other words, the perfection R(A) of A is a perfect ring of characteristic p together with a map θ : R(A) → A such that for any perfect ring B of characteristic p equipped with a map φ : B → A, there is a unique map f : B → R(A) such that φ factors through θ (i.e. φ = θf). The perfection of A may be constructed as follows. Consider the projective system
where the transition maps are the Frobenius endomorphism. The inverse limit of this system is R(A) and consists of sequences (x0, x1, ... ) of elements of A such that for all i. The map θ : R(A) → A sends (xi) to x0.[5]
See also
Notes
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References
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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 - 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 - 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 - 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 - 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
External links
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- ↑ Template:Harvnb, Section II.4
- ↑ Matsumura, Theorem 26.2
- ↑ Template:Harvnb
- ↑ Template:Harvnb, Section V.5.1.4, page 111
- ↑ Template:Harvnb, section 4.2