Integer
See Number.
Comments
An integer is an element of the ring of integers 
. The ring 
is the minimal ring which extends the semi-ring of natural numbers 
, cf. Natural number. Cf. Number for an axiomatic characterization of 
.
In algebraic number theory the term integer is also used to denote elements of an algebraic number field that are integral over 
. I.e. if 
is an algebraic field extension, where 
is the field of rational numbers, the field of fractions of 
, then the integers of 
are the elements of the integral closure of 
in 
, cf. Integral extension of a ring.
The integers of the algebraic number field 
, 
, are the elements 
, 
. They are called the Gaussian integers.
Let 
be a prime number. A 
-adic integer is an element of 
, the closure of 
in the field 
of 
-adic numbers. The field 
is the topological completion of the field 
for the 
-adic topology on 
which is defined by the non-Archimedean norm
 |
where 
if 
divides 
and 
does not divide 
, and 
.
References
| [a1] | Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1975) (Translated from Russian) (German translation: Birkhäuser, 1966) |
Integer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integer&oldid=33774