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See Number.


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 $k$.

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 $|0|_p=0$.


[a1] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966)
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