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ehrenfest
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[SOLVED] extension fields
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]
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?
Homework Statement
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]
Homework Equations
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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