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Consider the field $k$ obtained as the union of all finite towers of degree $2$ extensions over the rationals. Then $k$ has no degree $2$ extensions, yet $k$ admits extensions of every other finite degree.

A similar construction should hold in greater generality. Is there a name for such a field in the literature? The degree $2$ example reminds me of an analogue of the constructible numbers over $\mathbb{C}$.

Is there any reason that this construction fails in general?

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up vote 6 down vote accepted

This construction -- the compositum of all finite $p$-extensions of the base field $k$ within a fixed algebraic closure -- is called the $p$-closure of a field (at least by some -- the term does not appear to be particularly universal). And so no, there's no issues with the general construction.

One of the nicest things about it is that the cohomology of its Galois group is extraordinarily explicit, stemming largely from the fact that the $p$-th power map on $k^\times$ is necessarily surjective, at least when $k$ contains the $p$-th roots of unity (if it were not, adjoining a $p$-th root would give a further $p$-extension). This probably explains why you also only see the example for $p=2$ when taking $k=\mathbb{Q}$. The $p$-closure of a field turns out to be relatively crucial in the general theory of $p$-extensions because of this (easy cohomology computations are typically hard to come by!). A good reference is Koch's Galois Theory of $p$-Extensions.

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