3   The p adic numbers  Recall that    p is the completion of   with respect to        p  On Example Sheet 1  we will show that    p is a field  We also show that        p extends to    p and the associated valuation is discrete   Definition 3 1  The ring of p adic integers   p is the valuation ring        p     x     p     x   p   1         Facts      p is a discrete valuation ring  with maximal ideal  p   p  and non zero ideals are given by  p n   p   Proposition    p is the closure of   inside    p  In particular     p is the completion of   with respect to        p   Proof  Need to show   is dense in    p  Note   is dense in    p  Since    p     p is open  we have that    p     is dense in    p  Now         p         x         x   p   1       a b       p   b         p         Thus it suffices to show   is dense in     p     Let a b       p    a   b      p   b  For n      choose y n     such that b y n   a   m o d p n    THen y n   a b as n       In particular     p is complete and        p is dense     Definition   Inverse limit   Let   A n   n   1   be a sequence of sets   groups   rings together with homomorphisms   n   A n   1   A n  transition maps   Then the inverse limit  of   A n   n   1   is the set   group   ring defined by      lim n       A n       a n   n   1       n   1   A n       a n   1     a n   n         Define the group   ring operation componentwise    Notation  Let    m   lim n       A n   A m denote the natural projection    The inverse limit satisfies the following universal property   Proposition 3 2   Universal property of inverse limits    Assuming that   B is a set   group   ring    n are homomorphisms   n   B   A n such that  commutes for all n  Then  there exists a unique homomorphism      B   lim n       A n such that    n         n   Proof  Define         B     n   1   A n b     n   1     n   b       Then   n     n     n   1 implies that      b     lim n       A n  The map is clearly unique  determined by   n     n      and is a homomorphism of sets   groups   rings     Definition 3 3   I adic completion    Let I   R be an ideal  R a ring   The I adic completion of R is the      R       lim R         I n     where R   I n   1   R   I n is the natural projection   Note that there exists a natural map  i   R   R   by the Universal property of inverse limits  there exist maps R   R   I n   We say R is I adically  complete  if it is an isomorphism   Fact    ker   i   R   R         n   1   I n   Let    K           be a non archimedean  valued field and      O K such that         1   Proposition 3 4  Assuming that   K is complete with respect to        Then    i  Then   O K   lim n       O K     n O K    O K is   adically complete     ii  Every  x   O K can be written uniquely as x     i   0 n a i   i  a i   A  where  A   O K is a set of coset representatives for   O K     O K   Proof   i  K is complete and  O K is closed  so  O K is complete   x     n   1     n O K impies v   x     n v       for all n  and hence x   0  Hence   O K   lim O       K     n O K is injective   Let     x n   n   1     lim O       K     n O K and for each n  let  y n   O K be a lifting of   x n   O K     n O K  Then  y n   y n   1     n O K so that v   y n   y n   1     n v         Thus   y n   n   1   is a Cauchy sequence in  O K   Let  y n   y   O K  Then y maps to   x n   n   1   in the   lim n       O K     n O K  Thus   O K   lim n       O K     n O K is surjective    ii  Exercise on Example Sheet 1     Corollary 3 5   i    p   lim           p n      ii  Every element  x     p can be written uniquely as      x     i   n   a i p i       with n      a i     0   1         1     Proof   i  It suffices by Proposition 3 4 to show that        p   p n   p       p n         Let   f n         p   p n   p be the natural map      ker   f n       x         x   p   p   n     p n         hence       p n       p   p n   p is injective   Let         p   p n   p and let  c     p be a lift  Since   is dense in    p  there exists x     such that  x   c   p n   p is open in    p  Then f n   x        hence       p n       p   p n   p is surjective    ii  It follows from Proposition 3 4  ii  to  p   n x     p for some n        Example     1 1   p   1   p   p 2   p 3