In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz in 1910 as a part of his work on field theory.
where runs over all prime numbers, and each is zero, a natural number or infinity. Sometimes is used instead of . If no and there are only a finite number of non-zero then we recover the positive integers. Slightly less intuitively, if all are , we get zero. Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide "infinitely often," by taking that prime's corresponding exponent to be the symbol .
There is no natural way to add supernatural numbers, but they can be multiplied, with . Similarly, the notion of divisibility extends to the supernaturals with if for all . The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining
With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number. We can also extend the usual -adic order functions to supernatural numbers by defining for each
Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems from finite group theory carry over exactly. They are used to encode the algebraic extensions of a finite field. They are also used implicitly in many number-theoretical proofs, such as the density of the square-free integers and bounds for odd perfect numbers.
Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.
- Brawley, Joel V.; Schnibben, George E. (1989). Infinite algebraic extensions of finite fields. Contemporary Mathematics. 95. Providence, RI: American Mathematical Society. pp. 23–26. ISBN 0-8218-5101-2. Zbl 0674.12009.
- Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs. 124. Providence, RI: American Mathematical Society. p. 125. ISBN 0-8218-4041-X. Zbl 1103.12002.
- Fried, Michael D.; Jarden, Moshe (2008). Field arithmetic. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 11 (3rd ed.). Springer-Verlag. p. 520. ISBN 978-3-540-77269-9. Zbl 1145.12001.