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?