Supernatural number

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File:Steinitz numbers svg.svg of the lattice of supernatural numbers; primes other than 2 and 3 are omitted for simplicity.]]

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{{cite journal |last=Steinitz |first=Ernst |authorlink=Ernst Steinitz |date=1910 |language=German |title=Algebraische Theorie der Körper |journal=Journal für die reine und angewandte Mathematik |jfm=41.0445.03 |issn=0075-4102 |volume=137 |pages=167–309 |doi=10.1515/crll.1910.137.167 |url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002167042}}{{rp|249-251}} in 1910 as a part of his work on field theory.

A supernatural number $\backslash omega$ is a formal product:

: $\backslash omega\; =\; \backslash prod\_p\; p^\{n\_p\},$

where $p$ runs over all prime numbers, and each $n\_p$ is zero, a natural number or infinity. Sometimes $v\_p(\backslash omega)$ is used instead of $n\_p$. If no $n\_p\; =\; \backslash infty$ and there are only a finite number of non-zero $n\_p$ then we recover the positive integers. Slightly less intuitively, if all $n\_p$ are $\backslash infty$, we get zero.{{Citation needed|date=January 2022}} Supernatural numbers extend beyond natural numbers by allowing the possibility of infinitely many prime factors, and by allowing any given prime to divide $\backslash omega$ "infinitely often," by taking that prime's corresponding exponent to be the symbol $\backslash infty$.

There is no natural way to add supernatural numbers, but they can be multiplied, with $\backslash prod\_p\; p^\{n\_p\}\backslash cdot\backslash prod\_p\; p^\{m\_p\}=\backslash prod\_p\; p^\{n\_p+m\_p\}$. Similarly, the notion of divisibility extends to the supernaturals with $\backslash omega\_1\backslash mid\backslash omega\_2$ if $v\_p(\backslash omega\_1)\backslash leq\; v\_p(\backslash omega\_2)$ for all $p$. The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining

: $\backslash displaystyle\; \backslash operatorname\{lcm\}(\backslash \{\backslash omega\_i\backslash \})\; \backslash displaystyle\; =\backslash prod\_p\; p^\{\backslash sup(v\_p(\backslash omega\_i))\}$

and

: $\backslash displaystyle\; \backslash operatorname\{gcd\}(\backslash \{\backslash omega\_i\backslash \})\; \backslash displaystyle\; =\backslash prod\_p\; p^\{\backslash inf(v\_p(\backslash omega\_i))\}$.

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 $p$-adic order functions to supernatural numbers by defining $v\_p(\backslash omega)=n\_p$ for each $p$.

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.Brawley & Schnibben (1989) pp.25-26

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