Proving C is the Smallest Constructible Numbers Subfield with a>0 Property

[Dragonfall]

Homework Statement

Show that the field C of constructible numbers is the smallest subfield of R with the property that a\in C, a>0 \Rightarrow \sqrt{a}\in C.

The Attempt at a Solution

Suppose there's a proper subfield of C' of C that has that property, then let a\in C-C'. Somehow I must show that a is actually in C. Perhaps repeated squaring?

 

[Hurkyl]

Maybe a different tactic would be useful?

Let C' be the smallest subfield with that property. Can you prove C is a subfield of C', and that C has that property?