Supernatural numbers

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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] in 1910 as a part of his work on field theory.

A supernatural number \omega is a formal product:

\omega = \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(\omega) is used instead of n_p. If no n_p = \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 \infty, 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 \omega "infinitely often," by taking that prime's corresponding exponent to be the symbol \infty.

There is no natural way to add supernatural numbers, but they can be multiplied, with \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 \omega_1\mid\omega_2 if v_p(\omega_1)\leq v_p(\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

\displaystyle \operatorname{lcm}(\{\omega_i\}) \displaystyle =\prod_p p^{\sup(v_p(\omega_i))}
\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. We can also extend the usual p-adic order functions to supernatural numbers by defining v_p(\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.[2] 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.

References[edit]

  1. ^ Steinitz, Ernst (1910). "Algebraische Theorie der Körper". Journal für die reine und angewandte Mathematik: 167–309. ISSN 0075-4102. JFM 41.0445.03. 
  2. ^ Brawley & Schnibben (1989) pp.25-26

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