Ring of integers

In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension. It is particularly important in number theory since it measures the size of an ideal of a complicated number ring in terms of an ideal in a less complicated ring. When the less complicated number ring is taken to be the ring of integers, Z, then the norm of a nonzero ideal I of a number ring R is simply the size of the finite quotient ring R/I.

Relative norm

Let A be a Dedekind domain with the field of fractions K and B be the integral closure of A in a finite separable extension L of K. (In particular, B is Dedekind then.) Let $\mathrm{Id}(A)$ and $\mathrm{Id}(B)$ be the ideal groups of A and B, respectively (i.e., the sets of fractional ideals.) Following Template:Harv, the norm map

$N_{B/A}:\mathrm{Id}(B)\to \mathrm{Id}(A)$

is a homomorphism given by

$N_{B/A}(\mathfrak{q})={\mathfrak{p}}^{[B/\mathfrak{q}:A/\mathfrak{p}]},\mathfrak{q}\in \mathrm{Spec}B,\mathfrak{q}|\mathfrak{p}.$

If $L,K$ are local fields, $N_{B/A}(\mathfrak{b})$ is defined to be a fractional ideal generated by the set $\{N_{L/K}(x)|x\in \mathfrak{b}\}.$ This definition is equivalent to the above and is given in Template:Harv.

For $\mathfrak{a}\in \mathrm{Id}(A)$, one has $N_{B/A}(\mathfrak{a}B)={\mathfrak{a}}^n$ where $n=[L:K]$. The definition is thus also compatible with norm of an element: $N_{B/A}(xB)=N_{L/K}(x)A.$

Let $L/K$ be a finite Galois extension of number fields with rings of integer ${\mathcal{𝒪}}_K\subset {\mathcal{𝒪}}_L$. Then the preceding applies with $A={\mathcal{𝒪}}_K,B={\mathcal{𝒪}}_L$ and one has

$N_{L/K}(I)={\mathcal{𝒪}}_K\cap \prod _{\sigma \in G}^{}\sigma (I)$

which is an ideal of ${\mathcal{𝒪}}_K$. The norm of a principal ideal generated by α is the ideal generated by the field norm of α.

The norm map is defined from the set of ideals of ${\mathcal{𝒪}}_L$ to the set of ideals of ${\mathcal{𝒪}}_K$. It is reasonable to use integers as the range for $N_{L/\mathbf{Q}}$ since Z has trivial ideal class group. This idea does not work in general since the class group may not be trivial.

Absolute norm

Let $L$ be a number field with ring of integers ${\mathcal{𝒪}}_L$, and $\alpha$ a nonzero ideal of ${\mathcal{𝒪}}_L$. Then the norm of $\alpha$ is defined to be

$N(\alpha )=\left[{\mathcal{𝒪}}_L:\alpha \right]=|{\mathcal{𝒪}}_L/\alpha |.$

By convention, the norm of the zero ideal is taken to be zero.

If $\alpha$ is a principal ideal with $\alpha =(a)$, then $N(\alpha )=|N(a)|$. For proof, cf. Marcus, theorem 22c, pp65ff.

The norm is also completely multiplicative in that if $\alpha$ and $\beta$ are ideals of ${\mathcal{𝒪}}_L$, then $N(\alpha \cdot \beta )=N(\alpha )N(\beta )$. For proof, cf. Marcus, theorem 22a, pp65ff.

The norm of an ideal $\alpha$ can be used to bound the norm of some nonzero element $x\in \alpha$ by the inequality

$|N(x)|\le {\left(\frac{2}{\pi }\right)}^{r_2}\sqrt{\left|{\mathrm{\Delta }}_L\right|}N(\alpha )$

where ${\mathrm{\Delta }}_L$ is the discriminant of $L$ and $r_2$ is the number of pairs of complex embeddings of $L$ into $\mathbf{C}$.

See also

• Dedekind zeta function

References

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  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.
  
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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.
  
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• 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