“Finite extensions of local fields”的意思、由来-开放百科全书

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Finite extensions of local fields

释义

Unramified extension
Totally ramified extension
See also
References

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 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

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 .

References

{{cite book | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=Local Fields | series=London Mathematical Society Student Texts | volume=3 | publisher=Cambridge University Press | year=1986 | isbn=0-521-31525-5 | zbl=0595.12006 }}
{{cite book | last=Weiss | first=Edwin | title=Algebraic Number Theory | publisher=Chelsea Publishing | edition=2nd unaltered | year=1976 | isbn=0-8284-0293-0 | zbl=0348.12101 }}

1 : Algebraic number theory

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Unramified extension

Totally ramified extension

See also

References