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)