Let <I>F</i> be a field of characteristic 0 or a finite field.  If <I>E</i> is the splitting field over <I>F</i> for some polynomial in <I>F</i>[<i>x</i>],
then the mapping from the set of subfields of <i>E</i> containing <i>F</i> to the set of subgroups of Gal(<I>E/F</I>)  given by <I>K</i> &#8594  Gal(<i>E/K</i>)
  is a one-to-one correspondence.  Furthermore, for any subfield <I>K</i> of <I>E</i> containing <I>F</i>,<br>

1. [<i>E:K</i>] =  |Gal(<i>E/K</i>)| and  [<i>K:F</i>] =   |Gal(<i>E/F</i>)|/|Gal(<i>E/K</i>|.
[The index of Gal(<i>E/K</i>)   in Gal(<i>E/F</i>)   equals the degree of <I>K</I> over <I>F</i>.]<BR>

2.      If <I>K</i> is the splitting field of some polynomial in <i>F</I>[<I>x</i>], then  Gal(<i>E/K</i>)    is a normal subgroup of Gal(<i>E/F</i>)  and  Gal(<i>K/F</i>)  is isomorphic to Gal(<i>E/F</i>)/Gal(<i>E/K</i>). <br>

3. <i>K = E</i><sub> Gal(<i>E/K</i>)</sub>.   [The fixed field of Gal(<i>E/K</i>)   is <i>K</i>.]<br>

4.      If <i>H</I> is a subgroup of   Gal(<i>E/F</i>),  then <i>H </i>= Gal(<i>E/E<sub>H</sub></i>).
  [The automorphism group of <i>E</i> fixing <I>E<sub>H</sub></i>  is <I>H</i>.]