Does the splitting field of $X^3-2$ have a real embedding?
Real embedding of the splitting field of $X^3-2$
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field-theory
galois-theory
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2Note that if $x \ne y$ are two distinct roots of $X^3-2$, and $t = \frac{x}{y}$, then $t^3 = \frac{2}{2} = 1$ and $t \ne 1$. – 2012-10-08
1 Answers
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No. Briefly, because in $\mathbb C$, it has $3$ distinct roots, so the splitting field contains these, but in $\mathbb R$ there is only one root (say, because of strict monotonicity of $x\mapsto x^3$), so no embedding can be possible, as all $3$ roots should still satisfy the polynomial, hence would go into the same element, $\sqrt[3]2$.