What is an example of isomorphic but not equal fields?
Solution
Let \(\omega\) be a primitive cube root of unity. The fields \(\QQ(\sqrt[3]2)\) and \(\QQ(\omega\sqrt[3]2)\) are isomorphic, because \(\sqrt[3]2\) and \(\omega\sqrt[3]2\) have the same minimal polynomial \(x^3-2\) over \(\QQ\), so both fields are isomorphic to \(\QQ[x]/(x^3-2)\). They are not equal inside \(\CC\): \(\QQ(\sqrt[3]2)\subset\RR\), while \(\omega\sqrt[3]2\notin\RR\). See P-3VHPO for the full argument.