Does the splitting field of $X^3-2$ have a real embedding?
- Anybody can ask a question
- Anybody can answer
- The best answers are voted up and rise to the top
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, $\sqrt2$.