Ramification of local fields

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In algebraic number theory, through completion, the study of ramification of a prime ideal can often be reduced to the case of local fields where a more detailed analysis can be carried out with the aid of tools such as ramification groups.

In this article, a local field is non-archimedean and has finite residue field.

Unramified extension

Let L / K {\displaystyle L/K} {\displaystyle L/K} be a finite Galois extension of nonarchimedean local fields with finite residue fields ℓ / k {\displaystyle \ell /k} {\displaystyle \ell /k} and Galois group G {\displaystyle G} {\displaystyle G}. Then the following are equivalent.

  • (i) L / K {\displaystyle L/K} {\displaystyle L/K} is unramified.
  • (ii) O L / p O L {\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}{\mathcal {O}}_{L}} {\displaystyle {\mathcal {O}}_{L}/{\mathfrak {p}}{\mathcal {O}}_{L}} is a field, where p {\displaystyle {\mathfrak {p}}} {\displaystyle {\mathfrak {p}}} is the maximal ideal of O K {\displaystyle {\mathcal {O}}_{K}} {\displaystyle {\mathcal {O}}_{K}}.
  • (iii) [ L : K ] = [ ℓ : k ] {\displaystyle [L:K]=[\ell :k]} {\displaystyle [L:K]=[\ell :k]}
  • (iv) The inertia subgroup of G {\displaystyle G} {\displaystyle G} is trivial.
  • (v) If π {\displaystyle \pi } {\displaystyle \pi } is a uniformizing element of K {\displaystyle K} {\displaystyle K}, then π {\displaystyle \pi } {\displaystyle \pi } is also a uniformizing element of L {\displaystyle L} {\displaystyle L}.

When L / K {\displaystyle L/K} {\displaystyle L/K} is unramified, by (iv) (or (iii)), G can be identified with Gal ⁡ ( ℓ / k ) {\displaystyle \operatorname {Gal} (\ell /k)} {\displaystyle \operatorname {Gal} (\ell /k)}, which is finite cyclic.

The above implies that there is an equivalence of categories between the finite unramified extensions of a local field K and finite separable extensions of the residue field of K.

Totally ramified extension

Again, let L / K {\displaystyle L/K} {\displaystyle L/K} be a finite Galois extension of nonarchimedean local fields with finite residue fields l / k {\displaystyle l/k} {\displaystyle l/k} and Galois group G {\displaystyle G} {\displaystyle G}. The following are equivalent.

  • L / K {\displaystyle L/K} {\displaystyle L/K} is totally ramified.
  • G {\displaystyle G} {\displaystyle G} coincides with its inertia subgroup.
  • L = K [ π ] {\displaystyle L=K[\pi ]} {\displaystyle L=K[\pi ]} where π {\displaystyle \pi } {\displaystyle \pi } is a root of an Eisenstein polynomial.
  • The norm N ( L / K ) {\displaystyle N(L/K)} {\displaystyle N(L/K)} contains a uniformizer of K {\displaystyle K} {\displaystyle K}.

See also

References