Finite extensions of local fields

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

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Let   be a finite Galois extension of nonarchimedean local fields with finite residue fields   and Galois group  . Then the following are equivalent.

  • (i)   is unramified.
  • (ii)   is a field, where   is the maximal ideal of  .
  • (iii)  
  • (iv) The inertia subgroup of   is trivial.
  • (v) If   is a uniformizing element of  , then   is also a uniformizing element of  .

When   is unramified, by (iv) (or (iii)), G can be identified with  , 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

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Again, let   be a finite Galois extension of nonarchimedean local fields with finite residue fields   and Galois group  . The following are equivalent.

  •   is totally ramified
  •   coincides with its inertia subgroup.
  •   where   is a root of an Eisenstein polynomial.
  • The norm   contains a uniformizer of  .

See also

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References

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  • 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.