The post by Kevin Chung reminded me of an interesting equation, proposed to me by a friend.

**Problem.** Prove that there no non-trivial rational solutions \(a,b,c\) to the equation
\[a+b\sqrt[3]{2}+c\sqrt[3]{4}=0.\]

**Solution.** If we multiplicate original equation by \(\sqrt[3]{2}\), we'll obtain two similar equations. Lets write them all down
\[\begin{cases} a+b\sqrt[3]{2}+c\sqrt[3]{4}=0 \\
a\sqrt[3]{2}+b\sqrt[3]{4}+2c=0 \\
a\sqrt[3]{4}+2b+c\sqrt[3]{2}=0 \end{cases}
\]

Now we may try to think backwards. Could possibly a system of linear equations with rational coefficients give an irrational solution? Applying Cramer's rule we see that it's not the case. However, our system clearly has a solution \((2,\sqrt[3]{2},\sqrt[3]{4})\). Which contradicts the assumption that \((a,b,c)\) are rational numbers (keep in mind that we throw trivial \((0,0,0)\) solution away).

Can you propose any generalizations? What features of the tuple \((1,\sqrt[3]{2},\sqrt[3]{4})\) does this relation imply? Have you noticed how this relation is similar to conditions of linear independence?

## Comments

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TopNewestIf I may provide a "better" solution...

We note that we are considering arbitrary elements of \(\mathbb{Q}(\sqrt[3]{2})\). Also note that the minimal polynomial of \(\sqrt[3]{2}\) over \(\mathbb{Q}\) is \(x^3-2\), which has degree 3. Thus, \([\mathbb{Q}(\sqrt[3]{2}):\mathbb{Q}]=3\), so we have that \(\mathbb{Q}(\sqrt[3]{2})\) is a vector space over \(\mathbb{Q}\) with basis \(\{1,\sqrt[3]{2},(\sqrt[3]{2})^2\}\). Thus, there cannot be a nontrivial solution to \(a+b\sqrt[3]{2}+c\sqrt[3]{4}=0\) because \(\{1,\sqrt[3]{2},(\sqrt[3]{2})^2\}\) is linearly independent. – Jacob Erickson · 3 years, 5 months ago

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– Nicolae Sapoval · 3 years, 5 months ago

Bravissimo, that is exactly what I expected -- a larger context! Any ideas how to broaden the result? Maybe some more general equations?Log in to reply

– Jacob Erickson · 3 years, 5 months ago

I don't follow what you're saying. How would you like me to generalize?Log in to reply

I think , there exist a very simple solution , Since \[ a+b\sqrt[3]{2}+c\sqrt[3]{4}=0 \implies a^3+2b^3+4c^3=6abc \] Also , WLOG we can assume \( a,b,c \in \mathbb{Z} \) Now we must have \(a\) even , this will then give \( b \) even , and further going , by FMI(Fermat method of infinite descent , we get contradiction. – Shivang Jindal · 3 years, 5 months ago

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– Nicolae Sapoval · 3 years, 5 months ago

Indeed very clever and beautiful solution!Log in to reply

What is Cramer's rule?? – Satvik Golechha · 3 years, 2 months ago

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Google for that because I don't know how to explain that easily. T__T – Samuraiwarm Tsunayoshi · 2 years, 10 months ago

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