  For $p$ a prime and $a\in\mathbb{Q}$, where $a$ is not a $p^n$-th power of
any rational number, the extension $\mathbb{Q}(w_n)/\mathbb{Q}$ where
$w_n=\root p^n \of a$ is separable but non-normal. The Hopf-Galois theory for
separable extensions was determined by Greither and Pareigis, and the specific
classification for radical extensions such as these by the author. In this work
we extend this theory to a certain class of profinite extensions, namely those
formed from the union of these $\mathbb{Q}(w_n)$. We construct a 'profinite'
Hopf algebra which acts, and show that it satisfies a generalization of a
result due to Haggenmuller and Pareigis on the structure of Hopf algebra forms
of group algebras.
