LOCAL FIELD

In mathematics, a 'local field' is a special type of field that has a non-trivial absolute value and which is a locally compact topological field with respect to this absolute value. There are two basic types of local field: those in which the absolute value is Archimedean and those in which it is non-Archimedean. In the first case, one calls the local field an 'archimedean local field', in the second case, one calls it a 'non-archimedean local field'. There is an equivalent definition of non-archimedean local field given below. Local fields arise naturally in number theory as completions of global fields.
The complete classification of local fields (up to isomorphism) is the following:

★ Archimedean local fields (characteristic zero): the real numbers 'R', and the complex numbers 'C'.

★ Non-archimedean local fields of characteristic zero: finite extensions of the ''p''-adic numbers 'Q'_''p'' .

★ Non-archimedean local fields of characteristic ''p'': finite extensions of the field of formal Laurent series 'F'_''q''((''T'')) over a finite field 'F'_''q''.

Contents

Non-Archimedean local fields

Examples

See also

References

Non-Archimedean local fields

For a non-archimedean local field $F$, the following objects are very important:

★ its 'ring of integers' $mathcal\{O\}$ which is its closed unit ball $\{ain\; F:\; |a|leq\; 1\}$ (it is compact),

★ the 'units' in its ring of integers $mathcal\{O\}^\; imes$ which is its unit sphere $\{ain\; F:\; |a|=\; 1\}$,

★ the unique prime ideal in its ring of integers $mathfrak\{m\}$ which is its open unit ball ,

★ its residue field $k=mathcal\{O\}/mathfrak\{m\}$ which is finite (since it is compact and discrete).
One often talks about the (discrete) 'valuation' of a non-archimedean local field. This is a map $v:F$
ightarrowmathbb{R}cup{infty} obtained as follows: there is a real number 0 < ''c'' < 1 such that
:$c^\{v(a)\}=|a|mbox\{\; for\; all\; \}ain\; F$.
One generally chooses ''c'' such that ''v'' surjects onto $mathbb\{Z\}cup\{infty\}$, and calls this the ''normalized'' valuation.
An equivalent definition of a non-archimedean local field is that it is a field that is complete with respect to a discrete valuation and whose residue field is finite.

Examples

# 'The ''p''-adic numbers': the ring of integers of 'Q'_''p'' is the ring of ''p''-adic integers 'Z'_''p''. Its prime ideal is ''p'''Z'_''p'' and its residue field is 'Z'/''p'''Z'. Every non-zero element of 'Q'_p can be written as ''u'' ''p''^''n'' where ''u'' is a unit in 'Z'_''p'' and ''n'' is an integer, then ''v''(''u'' ''p''ⁿ) = ''n'' for the normalized valuation.
# 'The formal Laurent series over a finite field': the ring of integers of 'F'_''q''((''T'')) is the ring of formal power series 'F'_''q''''T''. Its prime ideal is (''T'') (i.e. the power series whose constant term is zero) and its residue field is 'F'_''q''. Its normalized valuation is related to the (lower) degree of a formal Laurent series as follows: $vleft(sum\_\{i=-m\}^infty\; a\_iT^i$
ight) = -m (where ''a''_−''m'' is non-zero).
# The formal Laurent series over the complex numbers is ''not'' a local field. For example, its residue field is 'C'''T''/(''T'') = 'C', which is not finite.

See also

★ Local class field theory

★ Hasse principle

References

★ Milne, James, 'Algebraic Number Theory'.

★ Local Fields, , Jean-Pierre, Serre, Springer-Verlag, 1995, ISBN 0-387-90424-7