No, not an order of an element, but all roots of all orders. For each element a in F, you adjoin all roots of the element a. That is, adjoin to F all roots of the equations xn-a=0, for all n, and all a in F.
When F contains all roots of unity, it will be just the maximal abelian extension of F.
no subject
When F contains all roots of unity, it will be just the maximal abelian extension of F.