Hi,
The lemma above is from Iwasawa's Local Class Field Theory.
Could you confirm that my argument for M = F' is correct:
We see that $$\psi \mid_{M} \in Gal(M/F) \ \text{and} \ \psi \mid_{F'} \in Gal(F'/F)$$ and $$\mid Gal(M/F)\mid = n \ \text{and} \ \mid Gal(F'/F)\mid = \ n$$ That means $$\psi^n(x)=x, \ x \ \in M $$ and $$\psi^n(x)=x, \ x \ \in F' $$ So, when we consider a generic element $$k \in MF'$$ we see that $$\psi^n(k)=k$$ Since $$\psi \ \text{is a generator of} \ Gal(F''/F)$$ that implies $$\mid (Gal(F''/F) \mid \leq n$$ But we know that $$[F'':F] = [F'':M][M:F]$$ so $$[F'':M]=1 \ \text{and that implies} \ F''=F'M=M$$ and finally $$F' = M$$ Thank you