14   Higher Ramification Groups  Let L   K be a finite Galois extension of local fields  and    L   O L a uniformiser   Definition 14 1   s th ramification group    Let v L be a normalised valuation in  O L  For s         1  the s th ramification group is      G s   L   K           Gal   K     v L       x     x     s   1   x   O L         Remark   G s only changes at integers   G s  s         1 used to define upper numbering    Example     G   1   L   K     Gal   L   K   G 0   L   K           Gal   L   K         x     x mod   L   x   O L     ker   Gal   L   K     Gal   k L   k       I L   K     Note   For s       0      G s   L   K     ker   Gal   L   K     Aut   O L     L s   1 O L         hence G s   L   K   is normal in G   1          G s   G s   1       G   1   Gal   L   K         Theorem 14 2   i  For s   1      G s         G 0   v L         L       L     s   1          ii    n   0   G n     1      iii  Let s       0  Then there exists an injective group homomorphism      G s   G s   1   U L   s     U L   s   1       induced by           L     L  This map is independent of the choice of   L   Proof  Let K 0   L be a maximal unramified extension of K in L  Upon replacing K by K 0  we may assume that L   K is totally ramified    i  Theorem 13 8 implies   O L   O K     L    Suppose v L         L       L     s   1  Let  x   O L  then x   f     L     f   X     O K   X            x     x       f     L       f     L     f         L       f     L             L       L   g     L       for some  g   X     O K   X    using the fact that X n   Y n     X   Y     X n   1       Y n   1    Thus      v L       x     x     v L         L       L     v L   g     L         0   s   1        ii  Suppose     Gal   L   K        1  Then       L       L  because L   K     L   and hence v L         L       L        Thus     G s for some s   0 by  i     iii  Note  for     G s  s       0            L       L     L s   1 O L     hence            L     L   1     L s O L   U L   s         We claim         G s   U L   s     U L   s   1             L     L     is a group homomorphism with kernel G s   1  For         G s  let       L     u   L   u   O L     Then             L     L             L           L         L     L       u   u       L     L       L     L     But      u     u     L s   1 O L since     G s  Thus     u   u   U L   s   1   and hence              L     L         L     L         L     L mod U L   s   1         Hence   is a group homomorphism  Moreover       ker               G s         L       L mod   L s   1     G s   1       If   L     a   L is another uniformiser   a   O L    Then            L       L         a   a         L     L         L     L mod U L   s   1           Corollary 14 3  Gal   L   K   is solvable    Proof  By Proposition 13 11  Theorem 14 2 and Theorem 13 4  for s         1      G s   G s   1   a subgroup   Gal   k L   k   if s     1   k L         if s   0   k L       if s   1     Thus G s   G s   1 is solvable for s     1  Conclude using Theorem 14 2  ii      Let characteristic k   p  Then p     G 0   G 1   and   G 1     p n  Thus G 1 is the unique  since normal  Sylow p subgroup of  G 0   I L   K   Definition 14 4  G 1 is called the wild inertial group  and G 0   G 1 is called the tame quotient   Suppose L   K is finite separable  Say L   K is tamely ramified if characteristic k   e L   K  Otherwise it is wildly ramified   Theorem 14 5  Assuming that     K     p        L   K finite  D L   K           L   K      Then      L   K     e L   K   1  with equality if and only  if tamely ramified  In particular  L   K unramified if and only if   D L   K   O L   Proof  Example Sheet 3 shows   D L   K   D L   K 0   D K 0   K  Suffices to check 2 cases    i  L   K unramified  Then    gives that   O L   O K        for some      O L with k L   k   a      Let g   X     O K   X   be the minimal polynomial of    Since   L   K       k L   k    we have that g     X     k   X   is the minimal polynomial of a    g     X   separable and hence g             0   m o d     L  Theorem 12 8 implies   D L   K     g           O L    ii  L   K totally ramified  Say   L   K     e   O L   O K     L      L a root of      g   X     X e     i   0 e   1 a i X i   O K   X       is Eisenstein  Then      g       L     e   L e   1     e   1     i   1 e   1 i a i   L i   1   v L   e       Thus v L   y       L       e   1  Equality if and only if p   e     Corollary 14 6  Suppose L   K is an extension of number fields  Let  P   O L   P   O K      Then e   P         1 if and only if  P   D L   K   Proof  Theorem 12 9 implies   D L   K     P D L P   K    Then use e   P         e L P   K   and Theorem 14 5     Example  K     p    p n a primitive p n th root of unity   L     p     p n    The p n th cyclotomic polynomial is        p n   X     X p n   1   p   1     X p n   1   p   2         1     p   X         See Example Sheet 3     p n   X   irreducible  hence   p n   X   is the minimal polynomial of   p n    L     p is Galois  totally ramified of degree p n   1   p   1             p n   1 a uniformiser in  O L    O L     p     p n   1       p     p n     Gal   L     p             p n        abelian     m   m where   m     p n       p n m      v L     m               v L     p n m     p n     v L     p n m   1   1         Let k be maximal such that p k   m   1  Then   p n m   1 is a primitive p n   k th root of unity  and hence   p n m   1   1 is a uniformiser     in  L       p     p n m   1    Hence      v L     p n m   1   1     e L   L     e L     p e L       p     L     p     L       p     p n   1   p   1   p n   k   1   p   1     p k       Theorem 14 2  i  implies that   m   G i if and only if p k   i   1  Thus      G i           p n       i   0   1   p k       p n   p k   1   1   i   p k   1   1   k   i   1     1   p n   1   1   i