Local field

A field that is complete with respect to a discrete valuation and has finite residue field. The structure of a local field is well known: 1) if the characteristic of is , then is a finite extension of the field of -adic numbers (cf. -adic number); 2) if the characteristic of is greater than , then is isomorphic to the field of formal power series over a finite field . Such fields are called local, in contrast to global fields (finite extensions of the fields or ), 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 generalization of the concept of a local field. Namely, an -dimensional local field is a sequence of complete discrete valuation rings together with isomorphisms

where is the residue field and is the field of fractions of a ring . Moreover, must be finite. There exists a structure theory for -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 -dimensional local fields (in terms of algebraic -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 -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