1   Absolute values  Definition 1 1   Absolute value    Let K be a field  An absolute value  on K is a function         K       0 such that   i    x     0 if and only if x   0    ii    x y       x     y   for all x   y   K    iii    x   y       x       y     x   y   K  triangly inequality    We say   K           is a valued field   Example  K             with usual absolute value   a   i b     a 2   b 2  Write         for this absolute value   K any field  The trivial absolute value is        x       0 x   0 1 x   0     Although this is technically an absolute value  it is not useful or interesting  so should be ignored   Definition 1 2   p adic absolute value    Let K      and p be a prime  For 0   x      write x   p n a b  where   a   p     1    b   p     1  The p adic  absolute value  is defined to be        x   p     0 x   0 p   n x   p n a b     Verification    i  Clear   ii  Write y   p m c d  Then        x y   p     p m   n a c b d   p   p   m   n     x   p   y   p        iii  Without loss of generality  m   n  Then        x   y   p     p n a d   p m   n b c b d   p   p   n   max     x   p     y   p         An absolute value        on K induces a metric  d   x   y       x   y   on K  hence a topology on K   Definition 1 3   Place    Let                  be absolute values on a field K  We say        and          are equivalent  if they induce the same topology  An equivalence class of absolute values is called a place   Proposition 1 4  Assuming that                   are  non trivial  absolute values on K   Then  the following are equivalent    i        and          are equivalent    ii    x     1             1 for all x   K    iii  There exists c       0 such that     x   c           for all x   K   Proof   i     ii         x     1   x n   0 w r t         x n   0 w r t             x       1      ii     iii    Note      x   c     x       c log   x     log   x      Let a   K   such that    a     1  exists since        is non trivial   We need that   x   K        log   x   log   a     log   x     log   a           Assume that      log   x   log   a     log   x     log   a           Choose m   n      with n   0  such that      log   x   log   a     m n   log   x     log   a           Then we have     n log   x     m log   a   n log   x       m log   a         Hence    x n a m     1 and    x n a m     1  contradiction  Similarly for the case where      log   x   log   a     log   x     log   a            iii     i    Clear      Remark           2 on   is not an absolute value by our definition  Some authors replace the triangle inequality by        x   y         x         y         for some fixed         0   Definition 1 5   Non archimedean    An absolute value        on K is said  to be non archimedean  if it satisfies the ultrametric inequality         x   y     max     x       y           If        is not non archimedean  then it is archimedean   Example          on   is archimedean         p is a non archimedean absolute value   Lemma 1 6  Assuming that     K           is non archimedean  x   y   K    x       y    Then     x   y       y     Proof       x   y     max     x       y         y       and        y     max     x       x   y         x   y            Proposition 1 7  Assuming that     K           is non archimedean    x n   n   1   a sequence in K    x n   x n   1     0  Then    x n   n   1   is Cauchy  In particular  if K is in addition complete  then   x n   n   1   converges   Proof  For     0  choose N such that    x n   x n   1       for n   N  Then N   n   m        x n   x m         x n   x n   1           x n   1     x m               The  In particular  is clear     Example  p   5  construct sequence   x n   n   1   in   such that    i  x n 2   1   0   m o d 5 n     ii  x n   x n   1   m o d 5 n    Take x 1   2  Suppose we have constructed x n  Let x n 2   1   a 5 n and set x n   1   x n   b 5 n  Then     x n   1 2   1   x n 2   2 b x n 5 n   b 2 5 2 n   1   a 5 n   2 b x n 5 n   b 2 5 2 n     We choose b such  that a   2 b x n   0   m o d 5    Then we have x n   1 2   1   0   m o d 5 n   1    Now  ii  implies that   x n   n   1   is Cauchy  Suppose x n   l      Then x n 2   l 2  But  i  tells us that x n 2     1  so l 2     1  a contradiction  Thus              5   is not complete    Definition 1 8  The p adic numbers   p is the completion of   with respect to        p   Analogy with     Notation  As is usual when working with metric spaces  we will be using the notation      B   x   r       y   K     x   y     r   B     x   r       y   K     x   y     r       Lemma 1 9  Assuming that     K           is a non archimedean valued field  Then    i  If z   B   x   r    then B   z   r     B   x   r     so open balls don t have a centre    ii  If z   B     x   r   then B     x   r     B     z   r      iii  B   x   r   is closed    iv  B     x   r   is open   Proof   i  Let y   B   x   r    Then   x   y     r hence       z   y         z   x       x   y       max     z   x       x   y       r     Thus B   x   r     B   z   r      follows by symmetry    ii  Same as  i     iii  Let y   B   x   r    If z   B   x   r     B   y   r   then B   x   r     B   z   r     B   y   r   Hence y   B   x   r    Hence B   x   r     B   y   r          iv  If z   B     x   r    then B   z   r     B     z   r     B     x   r