Galois #3

i wrote some of the very-very basics of field extensions , in fact the examples help to  understand what an extension from a field to another is.field_extensions

galois#2

Consider the set of all the homorfisms of an extension of two fields , so we can try to configure how the set of this homorfisms is related to the field or roots of a given polynomial , the number of this set with the degree of the extension .In generall if  then , where is the degree of  the extension from F to E.

For more details click here:  gal2

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.