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= | In [[algebraic number theory]], a '''fundamental unit''' is a generator (modulo the [[root of unity|roots of unity]]) for the [[unit group]] of the [[Algebraic integer|ring of integers]] of a [[number field]], when that group has [[rank of an abelian group|rank]] 1 (i.e. when the unit group modulo its [[torsion subgroup]] is [[infinite cyclic]]). [[Dirichlet's unit theorem]] shows that the unit group has rank 1 exactly when the number field is a [[real quadratic field]], a [[complex cubic field]], or a [[totally imaginary number field|totally imaginary]] [[quartic field]]. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a '''fundamental system of units'''.<ref>{{harvnb|Alaca|Williams|2004|loc=§13.4}}</ref> Some authors use the term '''fundamental unit''' to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. {{harvnb|Neukirch|1999|loc=p. 42}}). | ||
==Real quadratic fields== | |||
For the real quadratic field <math>K=\mathbf{Q}(\sqrt{d})</math> (with ''d'' square-free), the fundamental unit ε is commonly normalized so that |ε| > 1. Then it is uniquely characterized as the minimal unit whose absolute value is > 1. If Δ denotes the [[discriminant of an algebraic number field|discriminant]] of ''K'', then the fundamental unit is | |||
:<math>\epsilon=\frac{a+b\sqrt{\Delta}}{2}</math> | |||
where (''a'', ''b'') is the smallest solution to<ref>{{harvnb|Neukirch|1999|loc=Exercise I.7.1}}</ref> | |||
:<math>x^2-\Delta y^2=\pm4</math> | |||
in positive integers. This equation is basically [[Pell's equation]] or the negative Pell equation and its solutions can be obtained similarly using the [[continued fraction]] expansion of <math>\sqrt{\Delta}</math>. | |||
== | Whether or not ''x''<sup>2</sup> − Δ''y''<sup>2</sup> = −4 has a solution determines whether or not the [[class group]] of ''K'' is the same as its [[narrow class group]], or equivalently, whether or not there is a unit of norm −1 in ''K''. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of <math>\sqrt{\Delta}</math> is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then ''K'' does not have a unit of norm −1. However, the converse does not hold as shown by the example ''d'' = 34.<ref>{{harvnb|Alaca|Williams|2004|loc=Table 11.5.4}}</ref> In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if ''D''(''X'') is the number of real quadratic fields whose discriminant Δ < ''X'' is not divisible by a prime congruent to 3 modulo 4 and ''D''<sup>−</sup>(''X'') is those who have a unit of norm −1, then<ref>{{harvnb|Stevenhagen|1993|loc=Conjecture 1.4}}</ref> | ||
:<math>\lim_{X\rightarrow\infty}\frac{D^-(x)}{D(x)}=1-\prod_{j\geq1\text{ odd}}\left(1-2^{-j}\right).</math> | |||
In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by Étienne Fouvry and Jürgen Klüners<ref>{{harvnb|Fouvry|Klüners|2010}}</ref> who show that the converse fails between 33% and 59% of the time. | |||
==Cubic fields== | |||
If ''K'' is a complex cubic field then it has a unique real embedding and the fundamental unit ε can be picked uniquely such that |ε| > 1 in this embedding. If the discriminant Δ of ''K'' satisfies |Δ| ≥ 33, then<ref>{{harvnb|Alaca|Williams|2004|loc=Theorem 13.6.1}}</ref> | |||
:<math>\epsilon^3>\frac{|\Delta|-27}{4}.</math> | |||
For example, the fundamental unit of <math>\mathbf{Q}(\sqrt[3]{2})</math> is <math>1+\sqrt[3]{2}+\sqrt[3]{2^2}</math> whose cube is ≈ 56.9, whereas the discriminant of this field is −108 and | |||
:<math>\frac{|\Delta|-27}{4}=20.25.</math> | |||
==Notes== | |||
{{reflist}} | |||
==References== | |||
*{{Citation | |||
| last=Alaca | |||
| first=Şaban | |||
| last2=Williams | |||
| first2=Kenneth S. | |||
| title=Introductory algebraic number theory | |||
| year=2004 | |||
| publisher=[[Cambridge University Press]] | |||
| isbn=978-0-521-54011-7 | |||
}} | |||
* {{cite book | author=Duncan Buell | title=Binary quadratic forms: classical theory and modern computations | publisher=[[Springer-Verlag]] | year=1989 | isbn=0-387-97037-1 | pages=92–93 }} | |||
*{{citation | |||
| last=Fouvry | |||
| first=Étienne | |||
| last2=Klüners | |||
| first2=Jürgen | |||
| title=On the negative Pell equation | |||
| year=2010 | |||
| journal=[[Annals of Mathematics]] | |||
| volume=2 | |||
| number=3 | |||
| pages=2035–2104 | |||
| mr=2726105 | |||
}} | |||
*{{Neukirch ANT}} | |||
*{{citation | |||
| last=Stevenhagen | |||
| first=Peter | |||
| title=The number of real quadratic fields having units of negative norm | |||
| year=1993 | |||
| journal=Experimental Mathematics | |||
| volume=2 | |||
| number=2 | |||
| pages=121–136 | |||
| mr=1259426 | |||
}} | |||
[[Category:Algebraic number theory]] | |||
Revision as of 16:50, 11 September 2013
In algebraic number theory, a fundamental unit is a generator (modulo the roots of unity) for the unit group of the ring of integers of a number field, when that group has rank 1 (i.e. when the unit group modulo its torsion subgroup is infinite cyclic). Dirichlet's unit theorem shows that the unit group has rank 1 exactly when the number field is a real quadratic field, a complex cubic field, or a totally imaginary quartic field. When the unit group has rank ≥ 1, a basis of it modulo its torsion is called a fundamental system of units.[1] Some authors use the term fundamental unit to mean any element of a fundamental system of units, not restricting to the case of rank 1 (e.g. Template:Harvnb).
Real quadratic fields
For the real quadratic field (with d square-free), the fundamental unit ε is commonly normalized so that |ε| > 1. Then it is uniquely characterized as the minimal unit whose absolute value is > 1. If Δ denotes the discriminant of K, then the fundamental unit is
where (a, b) is the smallest solution to[2]
in positive integers. This equation is basically Pell's equation or the negative Pell equation and its solutions can be obtained similarly using the continued fraction expansion of .
Whether or not x2 − Δy2 = −4 has a solution determines whether or not the class group of K is the same as its narrow class group, or equivalently, whether or not there is a unit of norm −1 in K. This equation is known to have a solution if, and only if, the period of the continued fraction expansion of is odd. A simpler relation can be obtained using congruences: if Δ is divisible by a prime that is congruent to 3 modulo 4, then K does not have a unit of norm −1. However, the converse does not hold as shown by the example d = 34.[3] In the early 1990s, Peter Stevenhagen proposed a probabilistic model that led him to a conjecture on how often the converse fails. Specifically, if D(X) is the number of real quadratic fields whose discriminant Δ < X is not divisible by a prime congruent to 3 modulo 4 and D−(X) is those who have a unit of norm −1, then[4]
In other words, the converse fails about 42% of the time. As of March 2012, a recent result towards this conjecture was provided by Étienne Fouvry and Jürgen Klüners[5] who show that the converse fails between 33% and 59% of the time.
Cubic fields
If K is a complex cubic field then it has a unique real embedding and the fundamental unit ε can be picked uniquely such that |ε| > 1 in this embedding. If the discriminant Δ of K satisfies |Δ| ≥ 33, then[6]
For example, the fundamental unit of is whose cube is ≈ 56.9, whereas the discriminant of this field is −108 and
![{\displaystyle \mathbf {Q} ({\sqrt[{3}]{2}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3cc8d178c5719c8d3c700365446bc6860e184f99)
![{\displaystyle 1+{\sqrt[{3}]{2}}+{\sqrt[{3}]{2^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9b0fec11b662ccdc54de5cfd62bd448e6d94b5)
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.
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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
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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 - 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 - 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 - Template:Neukirch ANT
- 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