Local field

A field that is complete with respect to a discrete valuation and has finite residue field. The structure of a local field $K$ is well known: 1) if the characteristic of $K$ is $0$, then $K$ is a finite extension of the field $\mathbb{Q}_p$ of $p$-adic numbers; 2) if the characteristic of $K$ is greater than $0$, then $K$ is isomorphic to the field $k((T))$ of formal power series over a finite field $k$. Such fields are called local, in contrast to global fields (finite extensions of the fields $\mathbf{Q}$ or $k(T)$), and are means for studying the latter. For cohomological properties of Galois extensions of local fields see [1], and also Adèle; Idèle; and Class field theory.

To construct a class field theory of multi-dimensional schemes one uses a generalisation of the concept of a local field. Namely, an $n$-dimensional local field is a sequence $O_0,\ldots,O_n$ of complete discrete valuation rings together with isomorphisms $$ k(O_i) \stackrel{\sim}{\rightarrow} K(O_{i+1}) $$ where $k$ is the residue field and $K$ is the field of fractions of a ring $O$. Moreover, $k(O_n)$ must be finite. There exists a structure theory for $n$-dimensional local fields (see [3]).

References

[1]	J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) MR0554237 Zbl 0423.12016
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986) MR0911121 Zbl 0645.12001 Zbl 0153.07403
[3]	A.N. Parshin, "Abelian coverings of arithmetic schemes" Soviet Math. Dokl. , 19 : 6 (1978) pp. 1438–1442 Dokl. Akad. Nauk SSSR , 243 (1978) pp. 855–858 MR0514485 Zbl 0443.12006

Comments

The concept of a local field is sometimes extended to include that of discretely valued fields with arbitrary residue fields. There is a class field theory for local fields with perfect residue fields in terms of a certain fundamental group [a1], [a2]. For an account of the class field theory of $n$-dimensional local fields (in terms of algebraic K-theory) see also [a3]–[a5].

References

[a1]	J.-P. Serre, "Sur les corps locaux à corps résiduel algébriquement clos" Bull. Soc. Math. France , 89 (1961) pp. 105–154 MR0142534 Zbl 0166.31103
[a2]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) pp. 648–674 ((Appendix: M. Hazewinkel, Classes de corps local)) MR1611211 MR0302656 MR0284446 Zbl 0223.14009 Zbl 0203.23401 Zbl 0134.16503
[a3]	K. Kato, "Class field theory and algebraic K-theory" M. Raynaud (ed.) T. Shioda (ed.) , Algebraic geometry , Lect. notes in math. , 1016 , Springer (1983) pp. 109–126
[a4]	K. Kato, "Vanishing cycles, ramification of valuations and class field theory" Duke Math. J. , 55 (1987) pp. 629–661 MR0904945 Zbl 0665.14005
[a5]	A.N. [A.N. Parshin] Paršin, "Local class field theory" Proc. Steklov Inst. Math. , 165 (1985) pp. 157–185 Trudy Mat. Inst. Steklov. , 165 (1984) pp. 143–170