Supernatural number

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

ω = ∏ p p n p , {\displaystyle \omega =\prod _{p}p^{n_{p}},}

where p {\displaystyle p} runs over all prime numbers, and each n p {\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 n p {\displaystyle n_{p}} are nonzero and no index n p {\displaystyle n_{p}} is ∞ {\displaystyle \infty } , we get the positive natural numbers; for example, 60 = 2 2 ⋅ 3 ⋅ 5 {\displaystyle 60=2^{2}\cdot 3\cdot 5} can be represented as

2 2 3 1 5 1 7 0 11 0 13 0 … {\displaystyle 2^{2}3^{1}5^{1}7^{0}11^{0}13^{0}\dots }

There are also many supernatural numbers with no natural counterpart, such as

2 1 3 1 5 1 7 1 11 1 13 1 … {\displaystyle 2^{1}3^{1}5^{1}7^{1}11^{1}13^{1}\dots }

and

2 1 3 ∞ 5 0 7 0 11 0 13 0 … {\displaystyle 2^{1}3^{\infty }5^{0}7^{0}11^{0}13^{0}\dots }

Sometimes the notation v p ( ω ) {\displaystyle v_{p}(\omega )} is used instead of n p {\displaystyle n_{p}} . v p ( ω ) {\displaystyle v_{p}(\omega )} can be seen as the p-adic valuation of ω {\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 ∏ p p n p ⋅ ∏ p p m p = ∏ p p n p + m p {\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 ω 1 ∣ ω 2 {\displaystyle \omega _{1}\mid \omega _{2}} if v p ( ω 1 ) ≤ v p ( ω 2 ) {\displaystyle v_{p}(\omega _{1})\leq v_{p}(\omega _{2})} for all p {\displaystyle p} . The notion of the least common multiple and greatest common divisor can also be generalized for supernatural numbers, by defining

lcm ⁡ ( { ω i } ) = ∏ p p sup ( v p ( ω i ) ) {\displaystyle \displaystyle \operatorname {lcm} (\{\omega _{i}\})\displaystyle =\prod _{p}p^{\sup(v_{p}(\omega _{i}))}}

and

gcd ⁡ ( { ω i } ) = ∏ p p inf ( v p ( ω i ) ) {\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. Supernatural numbers also arise in the classification of uniformly hyperfinite algebras.

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