# Constructible Numbers

1. Mar 12, 2007

### Dragonfall

1. The problem statement, all variables and given/known data

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

3. 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?

2. Mar 12, 2007

### Hurkyl

Staff Emeritus
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?