# Find all $a,b \in {\mathbb Q}$ such that $a + b \sqrt[3]{4}$ is constructible.

Find all $$a,b \in {\mathbb Q}$$ such that $$a + b \sqrt[3]{4}$$ is constructible.

For this question, clearly, since every rational number is constructible, the sum of two constructible numbers is constructible and if $$a$$ and $$b$$ are constructible numbers, then $$ab$$ is constructible. Here if we want to show $$a + b \sqrt[3]{4}$$ is constructible, $$a$$ must be constructible and $$b \sqrt[3]{4}$$ must be constructible. Since $$\sqrt[3]{4}$$ is not constructible, $$b$$ needs to be $$0$$, thus any rational $$a$$ and $$b=0$$ satisfy this.

While since all the theorem I mentioned is under "if" condition, but the proof is using "only if", is there any formal proof or other methods to solve this?

• As you note, your argument does not work the way you have presented it. Knowing $a, x$ constructible $\Rightarrow$ $a+x$ constructible does not immediately mean $a+x$ constructible $\Rightarrow$ $a,b$ constructible. However, we do know that if $a<x$ are constructible, then $x-a$ is constructible; and if $x>0$ and $y$ are constructible, then $y/x$ is constructible. These should allow you to properly formulate your argument. – Sambo Aug 6 '19 at 4:05