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Note that Frac $\mathbb{Z}((x)) \ne\mathbb{Q}((x))$.

As a result of Some questions about the ring Z((x)), we know that it is a Dedekind domain with uncountably many primes, each of which is of the form $$p^n + a_1q + a_2q^2 + \cdots$$ By the Weierstrass preparation theorem, the residue fields of $\mathbb{Z}((x))$ are precisely fields of the form $\mathbb{Q}_p(\alpha)$, where $\alpha$ has positive $p$-adic valuation.

Another question is:

Does the Tate elliptic curve over Frac $\mathbb{Z}((x))$ satisfy weak weak approximation? (this would imply that Frac $\mathbb{Z}((x))$ is Hilbertian) Does it have the Hilbert property (does it satisfy Hilbert's irreducibility theorem?)

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    $\begingroup$ Elliptic curves over number fields never satisfy weak weak approximation. Why do you expect something different to happen in your case? $\endgroup$
    – Daniel Loughran
    Oct 5, 2015 at 15:16
  • $\begingroup$ @DanielLoughran Hmm interesting... do you know of any good references for this? $\endgroup$
    – Will Chen
    Oct 5, 2015 at 15:32
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    $\begingroup$ This is a special case of a general result of Minchev. See for example Theorem 1.4.5 of cims.nyu.edu/~tschinke/princeton/papers/aim-html/harari/… (the statement says only that weak approximation fails, but it is clear from the proof that it also shows that weak weak approximation fails). Perhaps this proof could be adapted to your case. $\endgroup$
    – Daniel Loughran
    Oct 5, 2015 at 15:57

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This does not answer the question about the Tate curve, but the question in the title: The field ${\rm Frac}(\mathbb{Z}((x)))={\rm Frac}(\mathbb{Z}[[x]])$ is Hilbertian.

Namely, by a result of Weissauer (Satz 7.2 in [1]), the quotient field of a (generalized) Krull domain of dimension at least 2 is Hilbertian. The PID $\mathbb{Z}$ is a Krull domain, and if $R$ is a Krull domain, then so is $R[[x]]$. For a more direct and more general statement see for example Corollary B in [2].

[1]: Rainer Weissauer. Der Hilbertsche Irreduzibilitätssatz. J. reine angew. Math. 334 (1982), pp. 203–220. https://eudml.org/doc/152457

[2]: Arno Fehm and Elad Paran. Klein approximation and Hilbertian fields. J. reine angew. Math. 676 (2013), pp. 213—225. https://www.degruyter.com/view/journals/crll/2013/676/article-p213.xml

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