Let $K$ be a finite normal extension of $F$ such that there are no proper intermediate extensions of $K/F$. Show that $[K:F]$ is prime. Give a conterexample if $K$ is not normal over $F$.
If a normal $K/F$ has no intermediate extensions, then $[K : F]$ is prime
4
$\begingroup$
field-theory
galois-theory
-
2Do you know some about Galois correspondance ? – 2012-02-12
-
1@Lierre: the Galois correspondence is not necessary here. You can prove it just starting from the definition of $[K:F]$. – 2012-02-12
-
1@DamianSobota: Maybe (although I don't see any immediate solution without it) ! But if it is an exercise, we should guess what is the expected proof. That's why I asked. – 2012-02-12
-
2@Damian: really? This is quite amazing: could you please elaborate? – 2012-02-12