Question

Prove that the set of all constructible numbers is a field.

Short answer

The set of all constructible number is a field.

Step-by-step solution

Properties of subset of the field.

Since the set of constructible number, K is a subset of the field of , we have to show that following requirements of a subfield.

1. K is nonempty.
2. $a,b\in K\Rightarrow a-b\in K$
3. $a,b\in K\Rightarrow a/b\in Kifb\ne 0$

Showing set of all constructible numbers is a field.

The set K of constructible number is a subset of R , it is closed under addition and multiplication and contains inverse of its nonzero elements , then by Step 1^st , Kis a subfield of R .

Hence, the set of all constructible numbers is a field.