Are the constructible numbers algebraic?

Question

The definition I found for algebraic numbers was as follows (a's are integers): A real number is algebraic if it is a solution to some polynomial equation...$$a_nx^n+a_{n-1}x^{n-1}+...a_2x^2+a_1x+a_0=0 $$ We can also assert that any constructible number is the root of some polynomial degree $2^r$ $(r\ge0)$ with rational coefficients. Can there be a way to use this to prove that constructable numbers are algebraic?