For which fields $k$ with $k^\times$ not $p$-divisible, does there exist finite $l/k$ such that $l^\times$ is $p$-divisible? Is there a prime $p$ and a field $k$, not real closed, with $k^\times$ not $p$-divisible, such that there exists a finite extension $l/k$ such that $l^\times$ is $p$-divisible?
This question came up since I have proved a result in which I have the hypothesis that $l^\times$ is not $p$-divisible for any finite extension of $k$. If there are no such fields as described, then I can simplify this hypothesis to simply assuming that $k^\times$ is not $p$-divisible.
Of course $\mathbb{R}^\times$ is not $2$-divisible as $-1$ is not a square, while $\mathbb{C}^\times$ is divisible. However I was not able to produce essentially different examples, so I wonder whether there are any.
If $k$ is not real closed or algebraically closed, it has infinite index in its algebraic closure. However it is not clear to me whether $k$ can have finite index in the closure of $k$ under taking $p$-th roots.
Note that the field $k$ can not be the function field of a variety (other than a point) some field $k'$, because $k'(X_0,\dots, X_n)$ does not satisfy the property, since taking $p^n$-th roots of $X_0$ gives an infinite tower of field extensions. It is also clear that $k$ cannot be a number field, but it is less clear if it can be an algebraic extension of $\mathbb{Q}$ or an infinite extension of some function field.
 A: For odd $p$ there are no such fields: If $a\in k$ is not a $p$-th power, then for every $n$ the polynomial $X^{p^n}-a\in k[X]$ is irreducible (see for example Theorem 9.1 in Chapter VI of Lang's Algebra), hence if $l/k$ is any finite extension of degree $d$, and $p^n>d$, then this $a$ has no $p^n$-th root in $l$.
For $p=2$ one has for example all the formally real Euclidean fields: Take for example the field of constructible real numbers. It is not quadratically closed, but adjoining $\sqrt{-1}$ to it will make it quadratically closed. Not sure if that is an "essentially different" example now though.
A: The formally real Euclidean fields Arno Fehm has given for $p=2$ are the only examples. In Lam's book on quadratic forms, he proves on page 270 as a consequence of the Diller-Dress theorem that if $l$ is quadratically closed and $k/l$ is finite, then $k$ is either quadratically closed or Euclidean.
Therefore the only examples are with $p=2$ and $k$ a formally real Euclidean field.
References:
Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2.
