8   Global Fields  Definition 8 1   Global field    A global field  is a field which is either    i  An algebraic number field   ii  A global function field  i e  a finite extension of   p   t     Lemma 8 2  Assuming that     K           is a complete discretely valued field  L   K a finite Galois extension with absolute value        L extending          Then  for x   L and     Gal   L   K    we have         x     L     x   L   Proof  Since  x         x     L is another absolute value on L extending        on K  the result follows from uniqueness of        L     Lemma 8 3   Kummer s Lemma    Assuming that     K           a complete discretely valued field  f   X     K   X   a separable irreducible polynomial with roots   1         n   K sep  K sep is the separable closure of K       K sep with              1             i       for i   2       n   Then    1   K         Proof  Let L   K        L     L     1         n    Then L     L is a Galois extension  Let     Gal   L     L    We have                 1                   1               1       using Lemma 8 2  Hence       1       1  so   1   K           Proposition 8 4  Assuming that     F           is a complete discretely valued field  f   X       i   0 n a i X i   O K   X   a separable irreducible monic polynomial      K sep a root of f  Then  there exists     0 such that for any  g   X       i   0 n b i X i   O K   X   monic with   a i   b i       for all i  there exists a root   of g   X   such that K         K          Nearby polynomials define the same extensions    Proof  Let   1         n   K sep be the roots of f which are necessarily distinct  Then f       1     0  We choose   sufficiently small such that   g     1         f       1     2 and   f       1     g       1         f       1      Then we have   g       1         f       1     2     g       1     2  the equality is by Lemma 1 6     By Hensel s Lemma version 1 applied to the field K     1   there exists     K     1   such that g         0 and           1       g       1      Then       g       1         f       1         j   1 n     1     j         1     i       for i   2       n   Use      1     i     1 since   i integral   Since          1         1     i             i   using Lemma 1 6  we have that Kummer s Lemma gives that   1   K       and hence K     1     K           Theorem 8 5  Assuming that   K is a local field  Then  K is the completion of a global field   Proof  Case 1          is archimedean  Then   is the completion of    and   is the completion of     i    with respect to             Case 2          non archimedean  equal characteristic  Then K     q     t     is the completion of   q   t   with respect to the t adic valuation   Case 3          non archimedean mixed characteristic  Then  K     p        with   a root of a monic irreducible polynomial  f   X       p   X    Since   is dense in    p  we choose g   X         X   as in Proposition 8 4  Then K           with   a root of g   X    Since         dense in    p         K  and K is complete  we must have that K  is the completion of