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From Encyclopedia of Mathematics
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See Number.

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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)
How to Cite This Entry:
Integer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Integer&oldid=33774