7   Local Fields  Definition 7 1   Local field    Let    K           be a valued field  Then K is a local field  if it is complete and locally compact   Reminder  locally compact means for all x   K  there exists U open and V compact  such that x   U   V   Example    and   are compact    Proposition 7 2  Assuming that     K           is a non archimedean complete valued field  Then  the following are equivalent    i  K is locally compact   ii  O K is compact   iii  v is discrete and  k   O K   m is finite   Proof   i     ii    Let U   0 be a compact neighbourhood of 0  0   U   Z with U open  Z compact   Then there exists  x   O K such that  x O K   U  Since  x O K is closed   x O K is compact  Hence  O K is compact    x O K     x   1 O K is a homeomorphism     ii     i    O K compact implies  a   O K is compact for all a   K  So K is locally compact    ii     iii    Let x   m  and  A x   O K be a set of coset representatives for   O K   x O K  Then   O K     y   A y   x O K is a disjoint open cover  So A x is finite by compactness of  O K  So   O K   x O K is finite  hence   O K   m O K is finite o Suppose v is not  discrete  Then let x 1   x 2     such that      v   x 1     v   x 2         0       Then   x O K   x 2 O K   x 3 O K       O K  But   O K   x O K is finite so can only have finitely many subgroups  contradiction    iii     ii    Since  O K is a metric space  it suffices to prove  O K is sequentially compact  Let   x n   n   1   be a sequence in  O K  and fix      O K a uniformiser  Since     i O K     i   1 O K   k   O K     i O K is finite for all i    O K     O K         i O K   Since   O K     O K is finite  there exists   a 1   O K     O K and a subsequence   x n   n   1   such that x 1 n   a mod   for all n   Since   O K     2 O K is finite  there exists   a 2   O K     2 O K and a subsequence   x 2 n   n   1   of   x 1 n   n   1   such that  x 2 n   a 2   m o d     2 O K  Continuing  this  we obtain sequences   x i n   n   1   for i   1   2     such that   1      x   i   1   n   n   1    is a subsequence of   x i n   n   1     2    For any i  there exists   a i   O K     i O K such that x i n   a i mod   i for all n   Then necessarily a i   a i   1 mod   i for all i   Now choose y i   x i i  This defines a subsequence of   x n   n   1    Moreover  y i   a i   a i   1   y i   1 mod   i  Thus y i is Cauchy  hence converges by completeness     Example   i    p is a local field    ii    p     t     is a local field   More on inverse limits   Let   A n   n   1 a sequence of sets   groups   rings and   n   A n   1   A n homeomorphisms   Definition 7 3   Profinite topology    Assume A n is finite  The profinite topology  on  A     lim n       A n is the weakest topology on A such that   n   A   A n is continuous for all n  where A n is equipped with the discrete topology   Fact    A   lim n       A n with the profinite topology is compact  totally disconnected and Hausdorff   Proposition 7 4  Assuming that   K is a non archimedean local field  Then  under the isomorphism   O K   lim n       O K     n O K       O K a uniformiser    the topology on  O K coincides with the profinite topology   Proof  One checks that the sets      B       a     n O K   n       1   a   O K       is a basis of open sets in both topologies   For         clear   For profinite topology    O K   O K     n O K is continuous if and only if  a     n O K is open for all  a   O K     Goal  Classify all local fields   Lemma 7 5  Assuming that   K is a non archimedean local field  L   K a finite extension  Then  L is a local field   Proof  Theorem 6 1 implies that L is complete and discretely valued  Suffices to show  k L     O L   m L is finite  Let   1         n be a basis for L as a  K vector space         sup  sup norm  equivalent to       L implies that there exists r   0 such that      O L     x   L     x   sup   r         Take a   K such that   a     r  then      O L     i   1 n a   i O K   L       Then  O L is finitely generated as a module over  O K  hence k L is finitely generated over k     Definition 7 6   Equal characteristic    A non archimedean valued field    K           has equal characteristic  if characteristic   K     characteristic   k    Otherwise it has mixed characteristic   Example    p has mixed characteristic    Theorem 7 7  Assuming that   K is a non archimedean local field of equal characteristic p   0  Then  K     p n     t     for some n   1   Proof  K complete discretely valued  characteristic K   0  Moreover  k     p n is finite  hence perfect   By Theorem 5 6  K     p n     t         Lemma 7 8  Assuming that   K a field  Then  an absolute value        is non archimedean if and only if    n   is bounded for all n       Proof      Since     1     1      n       n    it suffices to show that   n   bounded for n   1  Then note that        n       1   1       1     1           Suppose   n     B for all n      Let x   y   K with   x       y    Then we have       x   y   m       i   0 m m i x i y m   i       i   0 m   m i x i y m   i       y   m B   m   1       Taking m th roots gives        x   y       y     B   m   1     1 m       The right hand side tends to   y   as m      hence        x   y       y     max     x       y             Theorem 7 9   Ostrowski s Theorem    Assuming that         is a non trivial absolute value on    Then         is equivalent to either the usual absolute value          or the p adic absolute value        p for some prime p   Proof  Case          is archimedean  We fix b   1 an integer such that   b     1  exists by Lemma 7 8    Let a   1 be an integer and write b n in base a      b n   c m a m   c m   1 a m   1       c 0     with 0   c i   a  c m   0  Let B   max 0   c   a   1     c      and then we have       b n       m   1   B max     a   m   1       b       n   log a b   1   B   1   n     1 max     a   log a b   1   m   log a b n     b     max     a   log a b   1       Then    a     1 and        b       a   log a b           Switching roles of a and b  we also obtain        a       b   log b a            Then      and         gives  using log a b   log b log a       log   a   log a   log   b   log b           0       Hence    a     a   for all a       1  hence    x       x        for all x       Case 2          is non archimedean  As in Lemma 7 8  we have   n     1 for all n      Since        is non trivial  there exists n       1 such that    n     1  Write n   p 1 e 1   p r e r decomposition into prime factors  Then    p     1  for some p     p 1       p r    Suppose    q     1 for some prime q  q   p  Write 1   r p   s q with r   s      Then     1     r p   s q     max     r p       s q       1     contradiction  Thus    p         1 and    q     1 for all primes q   p  Hence        is equivalent to        p     Theorem 7 10  Assuming that     K           is a non archimedean local field of mixed characteristic  Then  K is a finite extension of    p   Proof  K mixed characteristic implies that characteristic K   0  hence     K  K non archimedean implies that                     p for some prime p  Since K is complete     p   K  Suffices to show that  O K is finite as a    p module   Let      O K be a uniformiser  v a normalised valuation and set v   p     e  Then   O K   p O K   O K     e O K is finite since     i O K     i   1 O K   O K     O K  is finite  Since     p       p     O K   p O K we have   O K   p O K a finite dimensional vector space over   p   Let  x 1       x n   O K be coset representatives for   p basis of   O K   p O K  Then          i   1 n a i x i   a i     0       r   1         is a set of coset representatives for   O K   p O K  Let  y   O K  Proposition 3 4  ii  tells us that     y     i   0       j   1 n a i j x j   p i   a i j     0       p   1         j   1 n     i   0   a i j p i   x j     p     Hence  O K is finite over    p     On Example Sheet 2 we will show that if K is complete and archimedean  then K     or    In summary   If K a local field  then either    i  K     or    archimedean     ii  K     p n     t      non archimedean equal characteristic     iii  K a finite extension of    p  non archimedean mixed characteristic