I'm new here! I'm studying Galois correspondence and i decided to take confidence with lattice structures (which I've never studied before). I want to ask you this question... given a field $E$ and the group Aut$(E)$, imagine that $\Phi$ is a map from the lattice subgroups of Aut$(E)$ to the lattice subfields of $E$ such that $\Phi(H)=E^H$ for all $H\in$ Aut$(E)$.
Denoting with the symbol $H\vee K$ the smallest subgroup of Aut$(E)$ that contains both $H$ and $K$, and with $H\wedge K$ the biggest subgroup of Aut$(E)$ contained in both $H$ and $K$ (and using the same notation also for th lattice of subfields...), then is it true that $\Phi$ is an anti-homomorphism of lattice i.e. $\Phi(H\vee K)=\Phi(H)\wedge \Phi(K)$ and $\Phi(H\wedge K)=\Phi(H)\vee \Phi(K)$ ??
All I've done is this:
- If $K\subset H$ then $\Phi(H)\subset \Phi(K)$.
Proof: If $b\in\Phi(H)=E^H=\{a\in E\ |\ \sigma(a)=a\ $for all $\sigma\in H\}$, then $\sigma(b)=b$ per ogni $\sigma\in K$, hence $b\in \Phi(K)=E^K$.
- Hence if $H,K\subset H\vee K$ then $\Phi(H\vee K)\subset \Phi(H),\Phi(K)$, so for sure $\Phi(H\vee K)\subset \Phi(H)\wedge \Phi(K)$. What can we say about the inverse inclusion? Is it true or it is just my wrong supposition?
Thank you to every one that will answer