S-integer
As a simple example, let be a finite set of rational prime numbers. The rational integers
,
, relatively prime (cf. also Mutually-prime numbers), such that the set of prime divisors of
(possibly empty) is contained in
are the so-called
-integers (corresponding to the specific set
). Clearly, this is a subring
of
. Let
denote the group of units of
, i.e. the group of multiplicatively invertible elements of
(the
-units). Clearly, these are
and the rational numbers
in the prime decomposition of which only prime numbers from the set
appear.
These notions can be defined in a more sophisticated way, the advantage of which is that it can be generalized to the more general case of a number field. For this the notion of absolute value on a number field is needed. Unfortunately, there is no general agreement on the definition of this notion. Below, this "absolute value" is taken in the sense of a metric as in [a1], Chap. 1, Sect. 4; Chap. 4, Sect. 4; equivalently, an absolute value is a function , where
is a fixed, conveniently chosen positive real number
and
is a valuation, as defined and used in [a2], Chap. 1, § 2; Chap. 3 § 1, (cf. also Valuation, which gives a slightly different definition).
In the special case above, every rational prime number gives rise to a
-adic absolute value and all possible absolute values of
are (up to topological equivalence) the
-adic ones (non-Archimedean), denoted by
, and the usual absolute value (Archimedean), denoted by
. Let
denote the set of absolute values (more precisely, the set of equivalence classes of absolute values (i.e. places) of
; cf. also Place of a field). Thus, every element of this set is of the form
, where
is either a rational prime number or the symbol
. One now modifies the definition of the set
above as the subset of
containing the absolute values (i.e. places)
, where
. Then
and
.
Consider now the more general situation, where a number field is taken in place of
and its ring of integers
is taken in place of
. Let
be the set of absolute values of
(more precisely, the set of equivalence classes of absolute values, i.e. places, of
). These are divided into two categories, namely, the non-Archimedean ones, which are in one-to-one correspondence with the prime ideals (or, what is essentially the same, with the prime divisors) of
and the Archimedean ones, which are in one-to-one correspondence with the isomorphic embeddings
(complex-conjugate embeddings giving rise to the same absolute value). As before, let
be a finite subset of
containing all Archimedean valuations of
. Then, the set
of
-integers and the set
of
-units are defined exactly as in the case of rational numbers (see the definitions above), where now
is replaced by
.
Many interesting problems concerning the solution of Diophantine equations are reduced to questions about -integers of "particularly simple form" (e.g. linear forms in two unknown parameters), which are
-units, and then results are obtained by applying a variety of relevant results on
-integers and
-units.
References
[a1] | Z.I. Borevich, I.R. Shafarevich, "Number Theory" , Acad. Press (1966) (In Russian) |
[a2] | W. Narkiewicz, "Elementary and analytic theory of algebraic numbers" , PWN/Springer (1990) |
S-integer. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=S-integer&oldid=50485