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Let $p$ be a prime. The axioms of a field of characteristic $p$ is definable in first order logic and form a satisfiable theory $T$. Indeed, $T$ has arbitrarily large finite models and it also has an infinite model, i.e. an infinite field of characteristic $p$: $\mathbb{Z}_p(x)$, the field of quotients of the polynomial ring $\mathbb{Z}_p[x]$. This field is countably infinite. Since $T$ has an infinite model $T$ has models of arbitrarily large cardinality by the Upward Löwenheim Skolem theorem, i.e. there are fields of characteristic $p$ with arbitrarily large cardinality.

My question is, are there any explicit examples of uncountable fields of characteristic $p$?

Edit: As Eoin pointed out below one example can be obtained by considering fractions of polynomials over $\mathbb{Z}_p$ in uncountable many variables $x_i$, $i\in I$, $|I|>\omega$.

Can anyone think of any other examples?

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    $\begingroup$ Add uncountably many variables. $\endgroup$ – Eoin Nov 17 '15 at 14:34
  • $\begingroup$ @Eoin Nice, did not think of that! $\endgroup$ – Bronski Stenberg Nov 17 '15 at 14:38

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