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A [[Field|field]] that is complete with respect to a discrete valuation and has finite residue field. The structure of a local field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601301.png" /> is well known: 1) if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601302.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601303.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601304.png" /> is a finite extension of the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601305.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601306.png" />-adic numbers (cf. [[P-adic number|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601307.png" />-adic number]]); 2) if the characteristic of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601308.png" /> is greater than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l0601309.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013010.png" /> is isomorphic to the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013011.png" /> of formal power series over a finite field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013012.png" />. Such fields are called local, in contrast to global fields (finite extensions of the fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013013.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013014.png" />), and are means for studying the latter. For cohomological properties of Galois extensions of local fields see [[#References|[1]]], and also [[Adèle|Adèle]]; [[Idèle|Idèle]]; and [[Class field theory|Class field theory]].
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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 a field|characteristic]] of $K$ is $0$, then $K$ is a finite extension of the field $\mathbb{Q}_p$ of [[P-adic number|$p$-adic number]]s; 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 field]]s (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 [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013016.png" />-dimensional local field is a sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013017.png" /> of complete discrete valuation rings together with isomorphisms
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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
 
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$$
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013018.png" /></td> </tr></table>
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k(O_i) \stackrel{\sim}{\rightarrow} K(O_{i+1})
 
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$$
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013019.png" /> is the residue field and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013020.png" /> is the [[field of fractions]] of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013021.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013022.png" /> must be finite. There exists a structure theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013023.png" />-dimensional local fields (see [[#References|[3]]]).
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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 [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) {{MR|0554237}} {{ZBL|0423.12016}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986) {{MR|0911121}} {{ZBL|0645.12001}} {{ZBL|0153.07403}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|0514485}} {{ZBL|0443.12006}} </TD></TR></table>
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<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top"> J.-P. Serre, "Local fields" , Springer (1979) (Translated from French) {{MR|0554237}} {{ZBL|0423.12016}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top"> J.W.S. Cassels (ed.) A. Fröhlich (ed.) , ''Algebraic number theory'' , Acad. Press (1986) {{MR|0911121}} {{ZBL|0645.12001}} {{ZBL|0153.07403}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top"> 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 {{MR|0514485}} {{ZBL|0443.12006}} </TD></TR>
 +
</table>
  
  
  
 
====Comments====
 
====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 [[#References|[a1]]], [[#References|[a2]]]. For an account of the class field theory of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013024.png" />-dimensional local fields (in terms of algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013025.png" />-theory) see also [[#References|[a3]]]–[[#References|[a5]]].
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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 [[#References|[a1]]], [[#References|[a2]]]. For an account of the class field theory of $n$-dimensional local fields (in terms of [[algebraic K-theory]]) see also [[#References|[a3]]]–[[#References|[a5]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.-P. Serre, "Sur les corps locaux à corps résiduel algébriquement clos" ''Bull. Soc. Math. France'' , '''89''' (1961) pp. 105–154 {{MR|0142534}} {{ZBL|0166.31103}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) pp. 648–674 ((Appendix: M. Hazewinkel, Classes de corps local)) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> K. Kato, "Class field theory and algebraic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060130/l06013026.png" />-theory" M. Raynaud (ed.) T. Shioda (ed.) , ''Algebraic geometry'' , ''Lect. notes in math.'' , '''1016''' , Springer (1983) pp. 109–126</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Kato, "Vanishing cycles, ramification of valuations and class field theory" ''Duke Math. J.'' , '''55''' (1987) pp. 629–661 {{MR|0904945}} {{ZBL|0665.14005}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> 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</TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[a1]</TD> <TD valign="top"> J.-P. Serre, "Sur les corps locaux à corps résiduel algébriquement clos" ''Bull. Soc. Math. France'' , '''89''' (1961) pp. 105–154 {{MR|0142534}} {{ZBL|0166.31103}} </TD></TR>
 +
<TR><TD valign="top">[a2]</TD> <TD valign="top"> M. Demazure, P. Gabriel, "Groupes algébriques" , '''1''' , North-Holland (1971) pp. 648–674 ((Appendix: M. Hazewinkel, Classes de corps local)) {{MR|1611211}} {{MR|0302656}} {{MR|0284446}} {{ZBL|0223.14009}} {{ZBL|0203.23401}} {{ZBL|0134.16503}} </TD></TR>
 +
<TR><TD valign="top">[a3]</TD> <TD valign="top"> 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</TD></TR>
 +
<TR><TD valign="top">[a4]</TD> <TD valign="top"> K. Kato, "Vanishing cycles, ramification of valuations and class field theory" ''Duke Math. J.'' , '''55''' (1987) pp. 629–661 {{MR|0904945}} {{ZBL|0665.14005}} </TD></TR>
 +
<TR><TD valign="top">[a5]</TD> <TD valign="top"> 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</TD></TR>
 +
</table>

Revision as of 17:35, 20 December 2014

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
How to Cite This Entry:
Local field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_field&oldid=35752
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article