I have some troubles with this problem:
Let $E/F$ be a Galois extension with Galois group cyclic. Prove that two intermediate $B_1$ and $B_2$ satisfy $B_1\subseteq B_2$ if $[E:B_2]$ divides $[E:B_1]$.
Thanks
I have some troubles with this problem:
Let $E/F$ be a Galois extension with Galois group cyclic. Prove that two intermediate $B_1$ and $B_2$ satisfy $B_1\subseteq B_2$ if $[E:B_2]$ divides $[E:B_1]$.
Thanks
HINT: By the Fundamental Theorem of Galois Theory, the problem is equivalent to the following question about cyclic groups:
If $G$ is a cyclic group, and $H_1$ and $H_2$ are subgroups of $G$, then $H_2\subseteq H_1$ if and only if $|H_2|$ divides $|H_1|$.
One direction is Lagrange's Theorem. The other comes from knowing stuff about cyclic groups.