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1. The problem statement, all variables and given/known data

Prove that x^2-3 is irreducible over [tex]\mathbb{Q}(\sqrt{2})[/tex].

EDIT: it should be [tex]\mathbb{Q}(\sqrt[3]{2})[/tex]

2. Relevant equations

3. The attempt at a solution

So, we want to show that [itex]\sqrt{3}[/itex] and [itex]-\sqrt{3}[/itex] are not expressible as [itex]a +b(\sqrt[3]{2})^2 +c\sqrt[3]{2}[/itex]. We could take the sixth power of both sides, but then the equation would be just messy. What else can we do?

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