let be the subgroup leaving fixed .
Let be a normal extension (splitting and seperable) of , where .
( is a field of a set of all element fixed by .)
( is means a order of )
above the diagrams are equavalent.
this is just the definition of .
by the definition of , is fixed in . this show that .
we prove .
Let . Since is a normal extension of , we can find an automoriphsim of leaving fixed and mapping onto different a zero of . therefore, at least there exists a element in that move . this implies that ,
we think about . a relations of them is . we shall suppose that
and shall derive contradiction. As a finite seperable extension, for some , Let
then . Let the elements of be . and consider the polynomial
this function is of degree and symmetric expressions in the . because H being a group, then , these cofficeients are invariant under each isomporiphism . Hence has coefficients in . and some is , so .
Therefore, we have
since is a normal extension of , . and the defintion is . therefore, .
we prove . Let and are int the same left coset of . Let . . for , and then . Hence, the same left coset of induce the same element in the so .
we prove is a normal extension of . every extension of , , is separable over . is separable over . every isomorphism of onto a subfield of leaving fixed can be extended to an automorphism of , since is normal over . thus the automorphisms of induce all possible isomorphisms of onto a subfield of leaving fixed. so then is a splitting filed over . and hence is nomrmal over .
by property 2, is the fixed filed of , so for , if and only if for all , . so . this means and , that is
this is the condition that be a normal subgroup of
when is a normal extension of , there is a one to one correpsondence between left coset over and automorphism of . for , let be the automorphism of induced by . thus . the map given by
for is a homomorphism. and is onto . the kernel of is . by the fundamental isomorphism theorem,