3

Let $K$ and $K_0$ be two isomorphic fields and let $F$ and $F_0$ be two isomorphic fields such that $K / F$ and $K_0 / F_0$ are two extensions. I have furthermore that $K_0 / F_0$ is monogenic and his degree is $n \in \mathbb{N}$. Can I state that $[K : F] = n$? Thank you very much in advance.

joseabp91
  • 2,360
  • 3
    What does monogenic extension mean? Is it the same as simple? As in a single generating element. In other words $K/F$ is monogenic if and only if there exists an element $z\in K$ such that $K=F(z)$? – Jyrki Lahtonen Aug 06 '18 at 08:35
  • Yes, a monogenic extension means that it is finitely generated only by one element. – joseabp91 Aug 06 '18 at 08:37
  • 3
    Is there something missing about the relation of the extensions $K/F$ and $K_0/F_0$? Otherwise let $K=\Bbb{Q}(x)$, $F=\Bbb{Q}(x^6)$, $K_0=F_0=\Bbb{Q}(x^3)$. They are all isomorphic, $[K_0:F_0]=1$, $[K:F]=6$? – Jyrki Lahtonen Aug 06 '18 at 08:40
  • 1
    -1 I see no effort or thought; even though counterexamples are so plentiful and easy to construct, especially in view of your previous question. – Servaes Aug 06 '18 at 09:39

0 Answers0