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{{DISPLAYTITLE:''p''-adic order}}
In [[number theory]], for a given [[prime number]] {{mvar|p}}, the '''{{mvar|p}}-adic order''' or '''{{mvar|p}}-adic [[Valuation (algebra)|additive valuation]]''' of a non-zero integer {{mvar|n}} is the highest [[exponent]] ν such that {{mvar|p}}<sup>ν</sup> [[divides]] {{mvar|n}}. The {{mvar|p}}-adic valuation of <math>0</math> is defined to be <math>\infty</math>. It is commonly abbreviated ν<sub>{{mvar|p}}</sub>({{mvar|n}}). If {{mvar|n/d}} is a rational number in lowest terms, so that {{mvar|n}} and {{mvar|d}} are relatively prime, then ν<sub>{{mvar|p}}</sub>({{mvar|n/d}}) is equal to ν<sub>{{mvar|p}}</sub>({{mvar|n}}) if {{mvar|p}} divides {{mvar|n}}, or -ν<sub>{{mvar|p}}</sub>({{mvar|d}}) if {{mvar|p}} divides {{mvar|d}}, or to 0 if it divides neither one. The most important application of the ''p''-adic order is in constructing the field of [[p-adic number|''p''-adic numbers]]. It is also applied toward various more elementary topics, such as the distinction between [[singly and doubly even]] numbers.<ref>
{{cite book
| author = David S. Dummit
| coauthors = Richard M. Foote
| year = 2003
| title = Abstract Algebra
| edition = 3rd
| publisher = Wiley
| isbn = 0-471-43334-9
}}</ref>
[[Image:2adic12480.svg|thumb|right|200px|Distribution of natural numbers by their 2-adic order, labeled with corresponding [[power of two|powers of two]] in decimal. Zero always has an infinite order]]


==Definition and Properties==
===Integers===
Let {{math|<VAR>p</VAR>}} be a prime in '''Z'''. The '''{{math|<VAR>p</VAR>}}-adic order''' or '''{{math|<VAR>p</VAR>}}-adic valuation''' for '''Z''' is defined as<ref>Ireland, K., Rosen, M. (2000). ''A Classical Introduction to Modern Number Theory''. Springer-Verlag New York. Inc., p. 3</ref> <math>\nu_p:\textbf{Z} \to \textbf{N}</math>


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:<math>
\nu_p(n)=
\begin{cases}
\mathrm{max}\{v\in\mathbb{N}:p^v | n\} & \text{if } n \neq 0\\
\infty & \text{if } n=0
\end{cases}
</math>
 
===Rational Numbers===
The {{math|<VAR>p</VAR>}}-adic order can be extended into the rational numbers. We can define<ref>Khrennikov, A., Nilsson, M. (2004). ''P-adic Deterministic and Random Dynamics''., Kluwer Academic Publishers, p. 9</ref> <math>\nu_p:\textbf{Q} \to \textbf{Z}</math>
 
:<math>
\nu_p\left(\frac{a}{b}\right)=\nu_p(a)-\nu_p(b).
</math>
 
Some properties are:
:<math>\nu_p(m\cdot n)= \nu_p(m) + \nu_p(n)~.</math>
:<math>\nu_p(m+n)\geq \inf\{ \nu_p(m), \nu_p(n)\}.</math> Moreover, if <math>\nu_p(m)\ne \nu_p(n)</math>, then <math>\nu_p(m+n)= \inf\{ \nu_p(m), \nu_p(n)\}.</math>
 
==''p''-adic Norm==
From our definition of the {{math|<VAR>p</VAR>}}-adic order, we can define the {{math|<VAR>p</VAR>}}-adic norm. The {{math|<VAR>p</VAR>}}-adic norm of '''Q''' is defined as <math>|\quad|_p: \textbf{Q} \to \textbf{R}</math>  
 
:<math>
|x|_p =
\begin{cases}
p^{-\nu_p(x)} & \text{if } x \neq 0\\
0 & \text{if } x=0
\end{cases}
</math>
 
Some properties of the {{math|<VAR>p</VAR>}}-adic norm:
 
:<math>
\begin{align}
|a|_p \geq 0 & \quad \text{Non-negativity}\\
|a|_p = 0 \iff a = 0 & \quad \text{Positive-definiteness}\\
|ab|_p = |a|_p|b|_p & \quad \text{Multiplicativeness}\\
|a+b|_p \leq |a|_p + |b|_p & \quad \text{Subadditivity}\\
|a+b|_p \leq \max\left(|a|_p, |b|_p\right) & \quad \text{it is non-archimedean}\\
|-a|_p = |a|_p & \quad \text{Symmetry} 
\end{align}
</math>
 
A metric space can be formed on the set '''Q''' with a (non-archimedean, translation invariant) metric defined by <math>d:\textbf{Q}\times\textbf{Q} \to \textbf{R}</math>
:<math>d(x,y)=|x-y|_p.</math>
 
==See also==
*[[Multiplicity (mathematics)]]
*[[Ostrowski's theorem]]
 
==References==
{{reflist}}
 
[[Category:Number theory]]

Latest revision as of 01:27, 7 November 2013

In number theory, for a given prime number Template:Mvar, the Template:Mvar-adic order or Template:Mvar-adic additive valuation of a non-zero integer Template:Mvar is the highest exponent ν such that Template:Mvarν divides Template:Mvar. The Template:Mvar-adic valuation of 0 is defined to be . It is commonly abbreviated νTemplate:Mvar(Template:Mvar). If Template:Mvar is a rational number in lowest terms, so that Template:Mvar and Template:Mvar are relatively prime, then νTemplate:Mvar(Template:Mvar) is equal to νTemplate:Mvar(Template:Mvar) if Template:Mvar divides Template:Mvar, or -νTemplate:Mvar(Template:Mvar) if Template:Mvar divides Template:Mvar, or to 0 if it divides neither one. The most important application of the p-adic order is in constructing the field of p-adic numbers. It is also applied toward various more elementary topics, such as the distinction between singly and doubly even numbers.[1]

Distribution of natural numbers by their 2-adic order, labeled with corresponding powers of two in decimal. Zero always has an infinite order

Definition and Properties

Integers

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Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-adic order
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A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

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for Z is defined as[2] νp:ZN

νp(n)={max{v:pv|n}if n0if n=0

Rational Numbers

The Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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νp(ab)=νp(a)νp(b).

Some properties are:

νp(mn)=νp(m)+νp(n).
νp(m+n)inf{νp(m),νp(n)}. Moreover, if νp(m)νp(n), then νp(m+n)=inf{νp(m),νp(n)}.

p-adic Norm

From our definition of the Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

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This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-adic norm. The Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-adic norm of Q is defined as ||p:QR

|x|p={pνp(x)if x00if x=0

Some properties of the Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer.-adic norm:

|a|p0Non-negativity|a|p=0a=0Positive-definiteness|ab|p=|a|p|b|pMultiplicativeness|a+b|p|a|p+|b|pSubadditivity|a+b|pmax(|a|p,|b|p)it is non-archimedean|a|p=|a|pSymmetry

A metric space can be formed on the set Q with a (non-archimedean, translation invariant) metric defined by d:Q×QR

d(x,y)=|xy|p.

See also

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

  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
  2. Ireland, K., Rosen, M. (2000). A Classical Introduction to Modern Number Theory. Springer-Verlag New York. Inc., p. 3
  3. Khrennikov, A., Nilsson, M. (2004). P-adic Deterministic and Random Dynamics., Kluwer Academic Publishers, p. 9