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In Haskell programming language, Fractional typeclass effectively represents a normed field, and has Floating as a sub-typeclass. Floating is to define $\exp$ and trigonometric functions (and related constants and functions). "Types" that satisfies Floating include $ℝ$ and $ℂ$. I wonder there would be other field that defines $\exp$.

I would define $\exp$ as Taylor series:

$$ \exp x = \sum_{n=0}^\infty \frac{x^n}{n!} $$

In a field, since $n \in ℤ$, $x^n$ is well-defined, division is well-defined, and if integers are defined as repeated addition of unity, $n!$ is well-defined. The only ways to make $\exp x$ undefined would be divergence of the series, or a division by zero. Convergence and divergence is well-defined because the field is normed, making the $\epsilon$-definition of limit of number sequence applicable.

As a consequence, a notable non-example would be $\text{GF}(p^k)$ because $p ≡ 0$, hence $p! ≡ 0$, resulting in a division by zero.

Is there an example rather than $ℝ$ and $ℂ$?

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    $\begingroup$ $\exp$ is also defined for $\mathbb{Q}_p$, the field of $p$-adic numbers, for each $p$, although the disc of convergence is smaller there. $\exp$ of operators (e.g., matrices) can also be defined using a series, although these do not form a field, but rather a normed vector space. $\endgroup$ – André 3000 Jan 28 at 4:41
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I’m not sure precisely what you’re asking, but perhaps you want to know about the $p$-adic fields $\Bbb Q_p$ and their finite (and infinite, when complete) extensions. There, the exponential series satisfies the rules you desire it to, except that the domain of convergence is so small as to make the exponential much less useful. The domain of convergence is $$\left\{z:\> \mid z\mid_p\><\frac1{p^{1/(p-1)}}\right\}\,.$$ However, the logarithm in this context is much better, with a domain of convergence $\{z:\>\mid z\mid_p<1\}$, and unlike the complex case, the logarithm is a homomorphism into the additive structure, throughout its domain of convergence.

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