In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.^[1] Given such a field, an absolute value can be defined on it. 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.

Every local field is isomorphic (as a topological field) to one of 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: the field of formal Laurent series F_q((T)) over a finite field F_q.

Contents 1 Non-Archimedean local field theory 1.1 Examples 2 Induced absolute value 3 See also 4 Notes 5 References 6 Further reading

[ Non-Archimedean local field theory

For a non-archimedean local field F (with absolute value denoted by |·|), the following objects are very important:

• its ring of integers which is its closed unit ball (it is compact),
• the units in its ring of integers which is its unit sphere ,
• the unique prime ideal in its ring of integers which is its open unit ball ,
• its residue field 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 obtained as follows: there is a real number 0 < c < 1 such that

.

One generally chooses c such that v surjects onto , 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:

   (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.

[ Induced absolute value

Given a locally compact topological field K, an absolute value can be defined as follows. First, consider the additive group of the field. As a locally compact topological group, it has a unique (up to positive scalar multiple) Haar measure μ. The absolute value is defined so as to measure the change in size of a set after multiplying it by an element of K. Specifically, define |·| : K → R by^[2]

for any measurable subset X of K (with 0 < μ(X) < ∞). This absolute value does not depend on X nor on the choice of Haar measure (since the same scalar multiple ambiguity will occur in both the numerator and the denominator).

Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology.^[3] Explicitly, for a positive real number m, define the subset B_m of K by

Then, thee B_m make up a neighbourhood basis of 0 in K.

[ See also

• Local class field theory
• Hasse principle

[ Notes

1. ^ Page 20 of Weil 1995
2. ^ Page 4 of Weil 1995
3. ^ Corollary 1, page 5 of Weil 1995

[ References

• Serre, Jean-Pierre (1995). Local Fields. Springer-Verlag. ISBN 0-387-90424-7. 
• Weil, André (1995), Basic number theory, Classics in Mathematics, Berlin, Heidelberg: Springer-Verlag, ISBN 3-450-58655-5 

[ Further reading

• J.W.S. Cassels, Local fields (1986) Cambridge University Press, ISBN 0-521-31525-5.
• A. Frohlich, "Local fields", in J.W.S. Cassels and A. Frohlich (edd), Algebraic number theory, Academic Press, 1973. Chap.I
• Milne, James, Algebraic Number Theory.
• Schikhoff, W.H. (1984) Ultrametric Calculus