Galois #1 (FixG)

Let E a field , the group of all automorfisms is . If F is a subset of E , then one automorfism that leaves all the elements of F invariant (  , for every a ) , is called F-automorfism. The set of all F-automorfism is called   and is a sub-group of the group of .

Let G a sub-group of of the field E. We call the set of all the invariant elements of E according to G-elements , thus :

FixG = {a | a , and , for every }.

The set FixG is a field , sub-field of E and is called fixed field of the group G.The fixed field of G(E,F) contains F and does not necessarily is the same field. 

more on FIxG later , we will see some interesting applications with respect to the roots of a polynomial  over a field.