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


As a result of [Some questions about the ring Z((x))](https://mathoverflow.net/questions/194056/some-questions-about-the-ring-zx), 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?)



Is Frac $\mathbb{Z}((x))$ Hilbertian?
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>


