Supernatural number

This article may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (December 2022) (Learn how and when to remove this message)

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^{[1]: 249–251} in 1910 as a part of his work on field theory.

A supernatural number ${\displaystyle \omega }$ is a formal product:

$${\displaystyle \omega =\prod _{p}p^{n_{p}},}$$

where ${\displaystyle p}$ runs over all prime numbers, and each ${\displaystyle n_{p}}$ is either zero, a natural number or infinity. This can be thought of as the prime factorization of the supernatural number. In the special case where only finitely many values ${\displaystyle n_{p}}$ are nonzero and no index ${\displaystyle n_{p}}$ is ${\displaystyle \infty }$, we get the positive natural numbers; for example, ${\displaystyle 60=2^{2}\cdot 3\cdot 5}$ can be represented as $${\displaystyle 2^{2}3^{1}5^{1}7^{0}11^{0}13^{0}\dots }$$ Zero can be represented as $${\displaystyle 2^{\infty }3^{\infty }5^{\infty }7^{\infty }11^{\infty }13^{\infty }\dots }$$ There are also many supernatural numbers with no natural counterpart, such as $${\displaystyle 2^{1}3^{1}5^{1}7^{1}11^{1}13^{1}\dots }$$ and $${\displaystyle 2^{1}3^{\infty }5^{0}7^{0}11^{0}13^{0}\dots }$$ (which could also be written ${\displaystyle 2^{1}3^{\infty }}$).

Sometimes ${\displaystyle v_{p}(\omega )}$ is used instead of ${\displaystyle n_{p}}$. ${\displaystyle v_{p}(\omega )}$ can be seen as the p-adic valuation of ${\displaystyle \omega }$, and it agrees with the standard p-adic valuation on the natural numbers.

There is no natural way to add supernatural numbers, but they can be multiplied, with ${\displaystyle \prod _{p}p^{n_{p}}\cdot \prod _{p}p^{m_{p}}=\prod _{p}p^{n_{p}+m_{p}}}$. Similarly, the notion of divisibility extends to the supernaturals with ${\displaystyle \omega _{1}\mid \omega _{2}}$ if ${\displaystyle v_{p}(\omega _{1})\leq v_{p}(\omega _{2})}$ for all ${\displaystyle p}$. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining

$${\displaystyle \displaystyle \operatorname {lcm} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\sup(v_{p}(\omega _{i}))}}$$

and

$${\displaystyle \displaystyle \operatorname {gcd} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\inf(v_{p}(\omega _{i}))}}$$.

With these definitions, the gcd or lcm of infinitely many natural numbers (or supernatural numbers) is a supernatural number.

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.^[2]

Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.

• Profinite integer

• Brawley, Joel V.; Schnibben, George E. (1989). Infinite algebraic extensions of finite fields. Contemporary Mathematics. Vol. 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. Vol. 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. Vol. 11 (3rd ed.). Springer-Verlag. p. 520. ISBN 978-3-540-77269-9. Zbl 1145.12001.

• Planet Math: Supernatural number