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Inspired by this question (In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?), I'm wondering for what fields there exists a non-trivial homomorphism from the additive group to the multiplicative group. As noted in the question linked above, such homomorphisms exist for $\mathbb{R}$ and $\mathbb{C}$ and a simple argument (given in my answer to the above question) shows that they do not exist for finite fields. The same argument also implies that no such homomorphism exists for algebraic closures of finite fields.

Is it known when such homomorphisms exist in general? Are there any examples that are not isomorphic to subfields of $\mathbb{C}$?

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    The relevant term for this idea would be an exponential field, which is a field equipped with such a homomorphism. I don't know anything about them except the name. – Milo Brandt Feb 18 '15 at 03:15
  • Interesting! Too bad that page doesn't give any new examples. – Qudit Feb 18 '15 at 03:26
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    Although I don't understand all the details of the argument, the answer to "are there examples not isomorphic to subfields of $\Bbb C$" is almost certainly yes, because the constraints of an exponential field form a first order language with an infinite model, which must thus have models of every infinite cardinality. – Mario Carneiro Feb 18 '15 at 04:17

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