Finite extensions 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 be a finite Galois extension of nonarchimedean local fields with finite residue fields /k and Galois group G. Then the following are equivalent.

  • (i) L/K is unramified.
  • (ii) 𝒪L/𝔭𝒪L is a field, where 𝔭 is the maximal ideal of 𝒪K.
  • (iii) [L:K]=[:k]
  • (iv) The inertia subgroup of G is trivial.
  • (v) If π is a uniformizing element of K, then π is also a uniformizing element of L.

When L/K is unramified, by (iv) (or (iii)), G can be identified with Gal(/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 be a finite Galois extension of nonarchimedean local fields with finite residue fields l/k and Galois group G. The following are equivalent.

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

See also

References

  • Cassels, J.W.S. (1986). Local Fields. London Mathematical Society Student Texts. Vol. 3. Cambridge University Press. ISBN 0-521-31525-5. Zbl 0595.12006.
  • Weiss, Edwin (1976). Algebraic Number Theory (2nd unaltered ed.). Chelsea Publishing. ISBN 0-8284-0293-0. Zbl 0348.12101.