Does Being an Automorphism of L Imply Being an Automorphism of K?

[chapani]

let K be an extention field of L and L be an extenstion field of F.

(1) t(a)=a ,for all t e G(K,F)

(2) t(a)=a ,for all t e G(L,F)

where e means "belongs to" , G(K,F) means "set of all automorphisms of K

leaving every element of F fixed and similarly for G(L,F).

i would like to know is (2) implies (1) or (1) implies (2)?

i think (2) implies (1) but not sure.[i have used simple logic no.of automorphisms of K =< no.of auto. of L]

if not then please give answer with counter example,if there no relation between

them then explain with reasion.

thanks in advanced for help me

 

[fresh_42]

We have $F \subseteq L \subseteq K$ and an $F-$automorphism of $K$ is automatically an $F-$automorphism of $L$, so (1) implies (2). The opposite is not generally true.