Surjective exponentials for algebraically closed fields

Question

The existence of the exponential on $\mathbb{C}$ has a very basic, yet very strong consequence : $(\mathbb{C}^*,\cdot)$ is a quotient of $(\mathbb{C},+)$. This question is concerned with fields $K$ such that $K^*$ is a quotient of $K$ ; that is, with the existence of a surjective group morphism $K\to K^*$. I will refer to such morphisms as "exponentials" on $K$. (I know that the notion of exponential fields exists, but I only consider the surjective case.)

The existence of such a map is not benign : since the additive group $K$ is $q$-divisible for all $q\neq char(K)$, it implies that $K^*$ is also $q$-divisible. In characteristic $0$, this actually implies that $K$ is algebraically closed (I suspect that in characteristic $p$ this implies that $K$ is separably closed, but I'll focus on zero characteristic for now). EDIT : That was false, but it doesn't really matter since Eric's answer gives a construction when $K^*$ is divisible.

Question 1: It's a basic fact that algebraically closed fields of characteristic zero and with the same transcendance degree over $\mathbb{Q}$ are isomorphic. So $\mathbb{C}_p \simeq \mathbb{C}$ for all $p$, and thus $\mathbb{C}_p$ admits (at least one) exponential map. On the other hand, isomorphisms $\mathbb{C}\to \mathbb{C}_p$ are highly non-constructible objects, so this does not give us any clue about what an exponential on $\mathbb{C}_p$ may look like.

Can such an exponential map $\mathbb{C}_p\to \mathbb{C}_p^*$ be explicitly defined ? In particular, does it exist without the axiom of choice (or with only a weak version) ?

Question 2: If we put the axiom of choice back on :

For any countable cardinal $\kappa$, do algebraically closed fields of transcendance degree $\kappa$ over $\mathbb{Q}$ admit a surjective exponential ? (In particular, what about $\overline{\mathbb{Q}}$ ?)

We already know from $\mathbb{C}$ that this is true for $\kappa = 2^{\aleph_0}$ so since "algebraically closed fields of characteristic zero with a surjective exponential" may be countably axiomatized in first-order logic, Löwenheim-Skolem implies that there are models of all infinite cardinality, so in particular question 2 has a positive answer for all uncountable $\kappa$, and for some countable $\kappa$. But since all countable $\kappa$ give models of the same cardinality $\aleph_0$, we cannot use that remark to answer the question.

Answer 1

I have no idea how to answer question 1, but in the presence of the axiom of choice, everything is very simple. If $K$ is an algebraically closed field of characteristic $0$, then the additive group of $K$ is isomorphic to a direct sum of $|K|$ copies of $\mathbb{Q}$ (this is trivial when $K$ is uncountable; when $K$ is countable note that it must have infinite degree over $\mathbb{Q}$ to be algebraically closed). The torsion subgroup of the multiplicative group (i.e., the roots of unity) is $\mathbb{Q}/\mathbb{Z}$, and since $\mathbb{Q}/\mathbb{Z}$ is injective, the multiplicative group splits as a direct sum $\mathbb{Q}/\mathbb{Z}\oplus A$ for some group $A$. The group $A$ is torsion-free and divisible and hence a $\mathbb{Q}$-vector space, and its dimension is easily seen to be $|K|$ (this is trivial when $K$ is uncountable; when $K$ is countable note that the prime integers are an infinite linearly independent set).

Thus $K^*$ is a direct sum of $\mathbb{Q}/\mathbb{Z}$ and $|K|$ copies of $\mathbb{Q}$. There is clearly a surjective homomorphism from $K$ to this direct sum, since there is a surjective homomorphism $\mathbb{Q}\to\mathbb{Q}/\mathbb{Z}$ and $|K|$ is infinite. In fact, there is a surjective homomorphism whose kernel is cyclic, just like in the case of $\mathbb{C}$.

Note that all that this argument needs from the axiom of choice is for $K$ to be well-orderable. So if you know $K$ is well-orderable without the axiom of choice, you get a surjective exponential without choice. In particular, for instance, $\overline{\mathbb{Q}}$ has a surjective exponential which you can actually write down (in terms of explicit bases for $\overline{\mathbb{Q}}$ and $\overline{\mathbb{Q}}^*$), though this will of course still not be very "natural".

Note also that it is not actually true that $K^*$ being divisible implies $K$ is algebraically closed. Indeed, this just says $K$ is closed under taking radicals, but since not all polynomials can be solved by radicals, this will not generate all roots of polynomials. The above discussion, however, applies perfectly well to any field of characteristic $0$ such that $K^*$ is divisible (with the modification that the group of roots of unity might be a subgroup of $\mathbb{Q}/\mathbb{Z}$ rather than all of $\mathbb{Q}/\mathbb{Z}$; however, as a divisible subgroup, it will be a direct summand of $\mathbb{Q}/\mathbb{Z}$, and hence still a quotient of $\mathbb{Q}$).

Finally, I would note that almost no field $K$ of characteristic $p>0$ has a surjective exponential. Indeed, every element of $K$ is annihilated by $p$, so this would mean $x^p=1$ for all $x\in K^*$. This implies $x=1$ for all $x\in K^*$, so the only such field is $K=\mathbb{F}_2$.