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)
[2]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)
[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

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
[a2]	M. Demazure, P. Gabriel, "Groupes algébriques" , 1 , North-Holland (1971) pp. 648–674 ((Appendix: M. Hazewinkel, Classes de corps local))
[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
[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