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See [[Number|Number]].
 
See [[Number|Number]].
  
 
====Comments====
 
====Comments====
An integer is an element of the ring of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512901.png" />. The ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512902.png" /> is the minimal ring which extends the semi-ring of natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512903.png" />, cf. [[Natural number|Natural number]]. Cf. [[Number|Number]] for an axiomatic characterization of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512904.png" />.
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An integer is an element of the ring of integers $\mathbf Z=\{\dots,-1,0,1,\dots\}$. The ring $\mathbf Z$ is the minimal ring which extends the semi-ring of natural numbers $\mathbf N=\{1,2,\dots\}$, cf. [[Natural number|Natural number]]. Cf. [[Number|Number]] for an axiomatic characterization of $\mathbf N$.
  
In algebraic number theory the term integer is also used to denote elements of an algebraic [[Number field|number field]] that are integral over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512905.png" />. I.e. if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512906.png" /> is an algebraic field extension, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512907.png" /> is the field of rational numbers, the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512908.png" />, then the integers of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i0512909.png" /> are the elements of the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129010.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129011.png" />, cf. [[Integral extension of a ring|Integral extension of a ring]].
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In algebraic number theory the term integer is also used to denote elements of an algebraic [[Number field|number field]] that are integral over $\mathbf Z$. I.e. if $k/\mathbf Q$ is an algebraic field extension, where $\mathbf Q$ is the field of rational numbers, the field of fractions of $\mathbf Z$, then the integers of $k$ are the elements of the integral closure of $\mathbf Z$ in $k$, cf. [[Integral extension of a ring|Integral extension of a ring]].
  
The integers of the algebraic number field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129013.png" />, are the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129015.png" />. They are called the Gaussian integers.
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The integers of the algebraic number field $\mathbf Q(i)$, $i^2+1=0$, are the elements $a+bi$, $a,b\in\mathbf Z$. They are called the Gaussian integers.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129016.png" /> be a prime number. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129018.png" />-adic integer is an element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129019.png" />, the closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129020.png" /> in the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129021.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129022.png" />-adic numbers. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129023.png" /> is the topological completion of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129024.png" /> for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129025.png" />-adic topology on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129026.png" /> which is defined by the non-Archimedean norm
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Let $p$ be a prime number. A $p$-adic integer is an element of $\mathbf Z_p$, the closure of $\mathbf Z$ in the field $\mathbf Q_p$ of $p$-adic numbers. The field $\mathbf Q_p$ is the topological completion of the field $\mathbf Q$ for the $p$-adic topology on $\mathbf Q$ which is defined by the non-Archimedean norm
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129027.png" /></td> </tr></table>
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$$\left|\frac ab\right|_p=p^{\nu_p(b)-\nu_p(a)},\quad a,b\in\mathbf Z,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129028.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129029.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129031.png" /> does not divide <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129032.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i051/i051290/i05129033.png" />.
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where $\nu_p(a)=r$ if $p^r$ divides $a$ and $p^{r+1}$ does not divide $a$, and $|0|_p=0$.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1975)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1975)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR></table>

Revision as of 06:47, 18 October 2014

See Number.

Comments

An integer is an element of the ring of integers $\mathbf Z=\{\dots,-1,0,1,\dots\}$. The ring $\mathbf Z$ is the minimal ring which extends the semi-ring of natural numbers $\mathbf N=\{1,2,\dots\}$, cf. Natural number. Cf. Number for an axiomatic characterization of $\mathbf N$.

In algebraic number theory the term integer is also used to denote elements of an algebraic number field that are integral over $\mathbf Z$. I.e. if $k/\mathbf Q$ is an algebraic field extension, where $\mathbf Q$ is the field of rational numbers, the field of fractions of $\mathbf Z$, then the integers of $k$ are the elements of the integral closure of $\mathbf Z$ in $k$, cf. Integral extension of a ring.

The integers of the algebraic number field $\mathbf Q(i)$, $i^2+1=0$, are the elements $a+bi$, $a,b\in\mathbf Z$. They are called the Gaussian integers.

Let $p$ be a prime number. A $p$-adic integer is an element of $\mathbf Z_p$, the closure of $\mathbf Z$ in the field $\mathbf Q_p$ of $p$-adic numbers. The field $\mathbf Q_p$ is the topological completion of the field $\mathbf Q$ for the $p$-adic topology on $\mathbf Q$ which is defined by the non-Archimedean norm

$$\left|\frac ab\right|_p=p^{\nu_p(b)-\nu_p(a)},\quad a,b\in\mathbf Z,$$

where $\nu_p(a)=r$ if $p^r$ divides $a$ and $p^{r+1}$ does not divide $a$, and $|0|_p=0$.

References

[a1] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966)
How to Cite This Entry:
Integer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integer&oldid=33774