An ordered field that is closed under the operation of taking square roots of all positive elements is called a Euclidean field; given any field $F$, there is a smallest Euclidean field containing $F$, called its Euclidean closure.
What if we want to generalize this notion to consider a field that is closed under the operation of taking all $n$th roots of all positive elements? That is, suppose $F$ is an ordered field with the property that for all $a>0$ and for every positive integer $n$, the equation $x^n = a$ has a solution.
- Is there a name for such a field?
- Certainly, $\mathbb R$ satisfies this condition; what is the smallest subfield of $\mathbb R$ with this property? It would seem to be countable, as one can obtain it by beginning with $\mathbb Q$ and adjoining all $n^{th}$ roots.