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MacLane (1939) calls a field $F$ universal if every other field $F'$ of the same cardinality and characteristic as $F$ is a subfield of $F$. He then exhibits an example, viz. a field of generalized formal power series.

Specifically, his Theorem 3 states that given

  • an ordered abelian group $\Gamma$ containing a nonzero element and such that for every $n \in \mathbb{Z}$ and $\alpha \in \Gamma$ there is some $\gamma \in \Gamma$ with $n\gamma = \alpha$, and
  • an algebraically closed field $\mathbb{K}$,

the field $\mathbb{K}\{t^\Gamma\}$ of all formal series of the form $\sum_{\alpha \in N \subseteq \Gamma} a_\alpha t^\alpha$ with $a_\alpha \in \mathbb{K}$ and $N$ well ordered by the linear order on $\Gamma$ is universal.

A Google scholar search shows that none of the papers citing this one mention "the field with one element" or "characteristic one". Yet it seems plausible that MacLane's definition can be suitably generalized, and that there is a universal semifield of characteristic one.

Is there an obvious way in which the notion of universality and the construction above inform $\mathbb{F}_1$, say by attempting to view $\mathbb{F}_1$ as a closure of some algebraic structure of generalized power series over the tropical semifield?

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  • $\begingroup$ what do you mean by "attempting to view it as a closure"? Do you want to define a new object? If yes, what are its properties you desire? $\endgroup$ Mar 7, 2017 at 9:16

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