15   Infinite Galois Theory  Definition 15 1   Infinite Galois definitions    L   K is separable if       L  the minimal polynomial f     X     K   X   for   is separable   L   K is normal if f     X   splits in L for all     L   L   K is Galois if it is separable and normal  Write Gal   L   K       Aut K   L   in this case  If L   K is a finite Galois extension  then we have a Galois correspondence        subextensions K   K     L       subgroups of Gal   L   K     K     Gal   K   K         Let   I       be a poset  Say I is a directed set if for all i   j   I  there exists k   I such that i   k  j   k   Example  Any total order  for example                  1 ordered by divisibility   Definition 15 2  Let   I       be a directed set and   G i   i   I a collection of groups together with maps   i j   G j   G i  i   j such that      i k     i j     j k for any i   j   k    i i   id  Say     G i   i   1     i j   is an inverse system  The inverse limit of   G i     i   is      lim i       G i       g i   i   I     i   I G i     i j   g j     g i         Remark             recovers the previous set   There exist projection maps   j   lim i     I     G i   G j   lim i     I     G i satisfies a universal property   Assume G i finite  Then the profinite topology on lim i     I     G i is the weakest topology such that   j are continuous for all j   I   Proposition 15 3  Assuming that   L   K Galois  Then    i  The set I     F   K finite   F   L   F Galois   is a directed set under      ii  For F   F     I  F   F   there is a restriction map res F   F     Gal   F     K     Gal   F   K   and the natural map      Gal   L   K     lim F     I     Gal   F   K       is an isomorphism   Proof  Example Sheet 4