The Natural Ring of Ordinals Has Prime Factorization

The natural ring of ordinals is a unique factorization domain—every number factors uniquely as a finite product of primes.

The natural ring of ordinals ⟨Ord⟩ is the mathematical system of numbers generated by the ordinals with the operations of natural sum and natural product, enabling not only addition and multiplication of these numbers, but also subtraction. For example, in this ring we may speak sensibly of the numbers:

\( \newcommand\bminus{\mathbin{\textbf{─}}}\newcommand\bplus{\mathbin{\textbf{+}}} \omega\bminus 7,\qquad ω^{\omega^2}\bplus\omega^{\omega+5}\bminus\omega^3,\qquad\text{ and }\qquad \omega^3\bminus\omega^{\omega^\omega}.\)

A ring is a certain kind of algebraic structure generalizing the familiar arithmetic of the integers ⟨ℤ,+,·⟩ to a more general or abstract realm of number objects. In ring theory, we aim to leverage our understanding of the integers to these more abstract realms. And indeed the natural ring of ordinals generalizes many features of the integers to a context including the transfinite ordinals. Today we shall how prime factorization is manifested in the natural ring of ordinals, with an ordinal version of the fundamental theorem of arithmetic—every number factors uniquely as a finite product of primes.

We introduced the basic construction of ⟨Ord⟩ in my previous essay, The Natural Ring of Ordinals, where we saw how the numbers of this ring can be represented as formal ordinal differences α ─ β, taken with respect to the same-difference equivalence relation. We outlined several attractive algebraic features to which this leads. The natural ring of ordinals, for example, is a discretely ordered cancellative commutative ring with identity. We provided several normal forms for representing the numbers in this ring, such as the signed Cantor normal form

\( \newcommand\bminus{\mathbin{\textbf{-\!-}}}\newcommand\bplus{\mathbin{\textbf{+}}}ω^{α_n}\cdot k_n\bplus\cdots\bplusω^{α_0}\cdot k_0,\)

where the coefficients k_i are taken from the integers, including negative integers. We saw the closely related normal form based on finite-support formal polynomial expressions ∑_β ω^β · k_β and another normal form based on finite signed sums of distinct powers of 2. The natural ring of ordinals, it turns out, is isomorphic to the subring of the surreal numbers generated by the ordinals, and so one may legitimately imagine these numbers, if desired, as surreal numbers. Meanwhile, the direct construction is simple and proceeds independently of any need for the surreal field.

We ended the previous essay on a cliff-hanger with several tantalizing questions left open, which I shall presently begin to take up in this essay:

• Does ⟨Ord⟩ admit a robust concept of prime numbers?

• Does ⟨Ord⟩ admit a robust concept of even and odd?

• Does ⟨Ord⟩ admit greatest common divisors?

• Is √2 irrational with respect to ⟨Ord⟩?

• What about √ω?

I shall aim to provide a general algebraic analysis of the natural ring of ordinals, showing that it is an integral domain and indeed, a unique factorization domain, which will be the key to several of the questions above. Next time, however, we will see that the natural ring of ordinals is not a Euclidean domain, nor a principal ideal domain, nor a Noetherian ring.

Our analysis here will be based on the extremely fruitful structural observation that the natural ring of ordinals is isomorphic to a vast discretely ordered polynomial ring over the integers, arising in a presentation as a transfinite tower of rings extending endlessly upward with newly created indefinite variables x_α, one for each ordinal α, with each new generator x_α added on top, larger in the order than every polynomial using only earlier variables.

\(\newcommand\bminus{\mathbin{\textbf{-\!-}}}\newcommand\bplus{\mathbin{\textbf{+}}} ℤ\ \subseteq \ ℤ[x_0]\ \subseteq \ ℤ\bigl[x_0,x_1\bigr]\ \subseteq \ \cdots\ \subseteq \ ℤ\bigl[x_α\bigr]_{α < λ}\ \subseteq \ \cdots\ \subseteq \ ℤ\bigl[x_α\bigr]_{α\in\textup{Ord}}\ \cong\ ⟨\textup{Ord}⟩. \)

Once we do this, it follows on general ring-theoretic grounds that ⟨Ord⟩ is a unique factorization domain and therefore supports a robust theory of irreducibility and prime numbers, by which every number factors uniquely as a finite product of primes. This observation in turn is the key to several of the questions asked above.

Let’s get into it!

You can find my whole essay series on the ordinals in the ordinals tag.