An infinite chain of fields
12 Sep 2022
About a year ago, I was considering the following question: Can the positive real numbers be given a field structure such that multiplication takes the role of addition and some other operation takes the role of multiplication? In other words, is there an operation \(* : \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+\) which makes \(\mathbb{R}_+ \setminus \{1\}\) into an abelian group and distributes over multiplication?
Although the answer seems simple to me now, I could not think of it last fall, because my mathematical knowledge was much more limited. It wasn’t until March that I realized that addition of real numbers is actually isomorphic to multiplication of positive real numbers (with \(\exp\) and \(\log\) as the isomorphisms). Therefore, any algebraic structure on the reals can be “carried over” to the positive reals, but with addition replaced by multiplication. This gives an easy solution to my question: \(x * y = \exp (\log x \cdot \log y)\). You can check that \((\mathbb{R}_+, \cdot, *)\) satisfies all the axioms of a field. For example, \(*\) has \(\exp 1\) as its identity, and the inverse of a number \(x\) is \(\exp \frac{1}{\log x}\).
One can apply this process again, defining \(x \dagger y = \exp (\log x * \log y) = \exp^2 (\log^2 x \cdot \log^2 y)\). With this definition, \((\mathbb{R}_{>1}, *, \dagger)\) is a field, with \(\exp 1\) as its “zero” element and \(\exp^2 1\) as its “one” element.
In fact, one can continue indefinitely, creating an infinite chain of fields \(\mathbb{R} = K_0 \supset K_1 \supset K_2 \supset \ldots\) , with multiplication in \(K_i\) equal to addition in \(K_{i+1}\). Explicitly, for each \(i \in \mathbb{N}\), define the following:
 \[K_i = \mathrm{Im} \exp^i = \{ \exp^i x \;  \; x \in \mathbb{R} \}\]

\(+_i : K_i \times K_i \to K_i\)
\(x +_i y = \exp^i (\log^i x + \log^i y)\)  \(\cdot_i : K_i \times K_i \to K_i\)
\(x \cdot_i y = \exp^i (\log^i x \cdot \log^i y)\)
With these definitions, the following properties all hold:
 \[x +_{i+1} y = x \cdot_i y\]

\(K_i\) is a field, with \(\exp^i 0\) as the “zero” element, \(\exp^i 1\) as the “one” element, \(\exp^i (\log^i x)\) as the “additive” inverse of \(x\), and \(\exp^i \frac{1}{\log^i x}\) as the “multiplicative” inverse of \(x\).
 \(\exp : K_i \to K_{i+1}\) and \(\log : K_{i+1} \to K_i\) are field isomorphisms.
In short, the exponential function gives rise to an infinite chain of abelian groups contained in the real numbers, with each operation distributing over the previous one!
The exponential numbers
When I discovered all of this back in March, I realized that one could extend the real numbers by continuing this chain in the opposite direction. But I didn’t work out the details until earlier today. It turns out that continuing the chain in both directions amounts to taking the colimit of the diagram
\[\ldots \overset{\exp}{\longleftarrow} \mathbb{R} \overset{\exp}{\longleftarrow} \mathbb{R} \overset{\exp}{\longleftarrow} \mathbb{R} \overset{\exp}{\longleftarrow} \ldots\]in the category of sets. One can construct this colimit (hereafter denoted \(E\)) by taking the quotient of \(\mathbb{R} \times \mathbb{Z}\) modulo the equivalence relation generated by \((x, i) \sim (\exp x, i  1)\). For each \(i \in \mathbb{Z}\), there is an embedding \(f_i : \mathbb{R} \to E\) given by \(f_i(x) = [x, i]\), where \([x, i]\) is the equivalence class of \((x, i)\). These functions satisfy \(f_i = f_{i1} \circ \exp\). Let \(K_i \subset E\) be the image of \(f_i\). Then there is an infinite descending chain: \(\ldots \supset K_{1} \supset K_0 \supset K_1 \supset \ldots\) .
Since the \(f_i\) are injective, each \(K_i\) can be given a natural field structure: \([x, i] +_i [y, i] = [x + y, i]\) and \([x, i] \cdot_i [y, i] = [x \cdot y, i]\). Moreover, addition in \(K_i\) is multiplication in \(K_{i1}\):
\[\begin{align*} & [x, i] +_i [y, i] \\ & = [x + y, i] \\ & = [\exp (x + y), i  1] \\ & = [\exp x \cdot \exp y, i  1] \\ & = [\exp x, i  1] \cdot_{i1} [\exp y, i  1] \\ & = [x, i] \cdot_{i1} [y, i] \\ \end{align*}\]Lastly, the exponential and logarithmic functions can be defined on \(E\) in a natural way: \(\mathrm{Exp} [x, i] = [x, i + 1]\) and \(\mathrm{Log} [x, i] = [x, i  1]\). For every \(i\), \(\mathrm{Exp}\) is a field isomorphism from \(K_i\) to \(K_{i+ 1}\), with \(\mathrm{Log}\) as its inverse. In some sense, \(E\) is a natural domain for the logarithmic function, since \(\mathrm{Log}\) is defined everywhere, or equivalently, every element of \(E\) is equal to \(\mathrm{Exp} \; x\) for some \(x\). With this in mind, in makes sense to refer to the elements of \(E\) as “exponential numbers”.