Wednesday, 1 April 2015

field theory - smallest normal extension containing an infinite algebraic extension

Let F/k be an algebraic extension. Let S(F/k) be the set of all embeddings of F over k into algebraic closure ka. I'm trying to prove that the smallest normal extension of k containing F is



E=σS(F/k)σF (Π is used to denote the compositum of fields)



In the finite extension case, matter is simple as if τ is embedding of E over k, στσ is an injective mapping of S(F/k) into S(F/k). Since S(F/k) is finite, the above mapping is bijective and E=σS(F/k)σF=τσF.




But in the infinite extension case when S(F/k) might not be finite, surjective part must be added to prove that τ induces permutation on S(F/k). I don't quite get an idea of how I should prove the surjectiveness.

No comments:

Post a Comment

real analysis - How to find limhrightarrow0fracsin(ha)h

How to find lim without lhopital rule? I know when I use lhopital I easy get $$ \lim_{h\rightarrow 0}...