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.

Non-Archimedean local fields

For a non-archimedean local field <math>F</math>, the following objects are very important:

• its ring of integers <math>\mathcal{O}</math> which is its closed unit ball <math>\{a\in F: |a|\leq 1\}</math> (it is compact),
• the units in its ring of integers <math>\mathcal{O}^\times</math> which is its unit sphere <math>\{a\in F: |a|= 1\}</math>,
• the unique prime ideal in its ring of integers <math>\mathfrak{m}</math> which is its open unit ball <math>\{a\in F: |a|< 1\}</math>,
• its residue field <math>k=\mathcal{O}/\mathfrak{m}</math> 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 <math>v:F\rightarrow\mathbb{R}\cup\{\infty\}</math> obtained as follows: there is a real number 0 < c < 1 such that

<math>c^{v(a)}=|a|\mbox{ for all }a\in F</math>.

One generally chooses c such that v surjects onto <math>\mathbb{Z}\cup\{\infty\}</math>, 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

1. The p-adic numbers: the ring of integers of Q_p is the ring of p-adic integers Z_p. Its prime ideal is pZ_p and its residue field is Z/pZ. Every non-zero element of Q_p can be written as u pⁿ where u is a unit in Z_p and n is an integer, then v(u pⁿ) = n for the normalized valuation.
2. 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: <math>v\left(\sum_{i=-m}^\infty a_iT^i\right) = -m</math> (where a_−m is non-zero).
3. 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.
• Serre, Jean-Pierre (1995). Local Fields. Springer-Verlag. ISBN 0-387-90424-7.