Exponential-like functions on fields other than $\mathbb R$

Question

Let $F$ be any field, and say that $\epsilon:F\to F^\times$ is an exponential-like function if it is a group homomorphism from the additive group $(F, +)$ to the multiplicative group $(F, \times)$. That is, $\epsilon$ should satisfy $$\epsilon(0_F)=1_F$$ $$\epsilon(a+b)=\epsilon(a)\epsilon(b)$$

Such a function generalizes the properties of exponential functions on $\mathbb R$. Note that for many fields, no nontrivial $\epsilon$ exists. For example:

1. If $F=\mathbb Z/(p)$ then we would need to have $1=\epsilon(0)=\epsilon(p\cdot 1)=\epsilon(1)^p$, so $\epsilon(1)$ would need to be an element $b$ satisfying $b^p=1$. But by Fermat's Little Theorem, every $b\in\mathbb Z/(p)$ satisfies $b^p = b$, and therefore the only possible $\epsilon$ is the trivial homomorphism given by $\epsilon(1)=1$.
2. If $F=\mathbb Q$ then no such $\epsilon$ can exist, because we would need $\epsilon(1/n) = \epsilon(1)^{1/n}\in\mathbb Q$ for all $n$, but the only rational number that has rational roots of all orders is $1$.
3. On the other hand, taking $F = \mathbb C$ we have the familiar complex exponential, $\epsilon(x + iy) = \cos(x) + i \sin(y)$, which has all of the desired properties.

I am interested in the general question of which fields admit such an $\epsilon$. It seems likely that any algebraically-closed field would work, although I'm not sure how to go about proving it. What about fields like $\mathbb R(x)$?

Answer 1

This is pretty simple because the structure of the additive group $F$ is very simple: it's just a big direct sum of copies of $\mathbb{Q}$ in characteristic $0$, or of $\mathbb{F}_p$ in characteristic $p$. So, there is a nontrivial homomorphism $F\to F^\times$ iff there is a nontrivial homomorphism $F_0\to F^\times$ where $F_0$ is the prime subfield.

In characteristic $p$, there is no nontrivial such homomorphism, since any element of its image is an element $a\in F$ such that $a^p=1$, and the only such element is $1$ (since $x^p-1$ factors as $(x-1)^p$).

In characteristic zero, there is a nontrivial homomorphism $\mathbb{Q}\to F^\times$ iff $F$ has an element besides $1$ which has an $n$th root for all $n$ (given such an element, you can choose a system of compatible $n!$th roots for each $n$ by König's infinity lemma and then map $1/n!$ to the chosen $n!$th root). In particular, such an element exists if $F$ is algebraically closed, for instance.

A rather different question is whether there is a natural or useful exponential function. The examples above are highly artificial, as they involve picking a basis for $F$ over the prime field (which often cannot be done without the axiom of choice). I don't know of examples besides subfields of $\mathbb{C}$ which have an exponential function which is good for anything. The $p$-adic numbers $\mathbb{Q}_p$ have a natural and useful "exponential function" (defined by the usual power series) but it is not defined on the entire field (only on the proper subset where the power series converges, which turns out to be $p\mathbb{Z}_p$ for $p\neq 2$ and $p^2\mathbb{Z}_p$ for $p=2$) and so it does not meet your definition.