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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?

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    $\begingroup$ 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. $\endgroup$ – Sambo Aug 6 '19 at 4:05

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