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This question is about the Remark on the top of page 22 of Serre's Topics in Galois Theory, available here :

http://www.ms.uky.edu/~sohum/ma561/notes/workspace/books/serre_galois_theory.pdf

My question is as follows.

Let $L/K$ be a finite separable extension of fields. Assume that $L$ is Hilbertian. Does it follows that $K$ is Hilbertian?

I believe the answer is yes (and this shouldn't be too hard to show). However, Serre claims that the answer is negative, and refers in his remark to Kuyk's paper

W. Kuyk. Extensions de corps hilbertiens, J. of Algebra 14 (1970), 112-124

However, Kuyk seems to only give an example of a field extension $L/K$ of infinite degree such that $K$ is not Hilbertian, but $L$ is. The example is given in the Remarque on page 2 of his article. Note that $K$ can be taken to be the maximal $2$-extension of $\mathbb{Q}$. This is not a Hilbertian field, whereas its extension $L$ given by the maximal nilpotent extension of $\mathbb{Q}$ (or $K$) is Hilbertian.

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The maximal (pro-)solvable extension $L$ of $\mathbb{Q}$ is not Hilbertian, but every proper finite extension of $L$ is. See Fried-Jarden (third edition, 2008), Example 13.9.5.

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    $\begingroup$ Might be worth clarifying that the $L$ in this answer is the $K$ in the question. $\endgroup$
    – Greg Martin
    Commented Nov 5, 2019 at 20:17

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