Local field

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 not. 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. 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 (where p is any prime number).
Non-archimedean local fields of characteristic p (for p any given prime number): the field of formal Laurent series F_q((T)) over a finite field F_q (where q is a power of p).

There is an equivalent definition of non-archimedean local field: it is a field that is complete with respect to a discrete valuation and whose residue field is finite. However, some authors consider a more general notion, requiring only that the residue field be perfect, not necessarily finite.^[2] This article uses the former definition.

1 Induced absolute value
2 Non-archimedean local field theory
3 Higher dimensional local fields
4 See also
5 Notes
6 References
7 Further reading

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^[3]

$|a|:=\frac{\mu(aX)}{\mu(X)}$

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.^[4] Explicitly, for a positive real number m, define the subset B_m of K by

$B_m:=\{ a\in K:|a|\leq m\}.$

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

Non-archimedean local field theory

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

its ring of integers $\mathcal{O} = \{a\in F: |a|\leq 1\}$ which is a discrete valuation ring, is the closed unit ball of F, and is compact;
the units in its ring of integers $\mathcal{O}^\times = \{a\in F: |a|= 1\}$ which forms a group and is the unit sphere of F;
the unique non-zero prime ideal $\mathfrak{m}$ in its ring of integers which is its open unit ball $\{a\in F: |a|< 1\}$ ;
a generator ϖ of $\mathfrak{m}$ called a uniformizer of F;
its residue field $k=\mathcal{O}/\mathfrak{m}$ which is finite (since it is compact and discrete).

Every non-zero element a of F can be written as a = ϖⁿu with u a unit, and n a unique integer. The normalized valuation of F is the surjective function v : F → Z ∪ {∞} defined by sending a non-zero a to the unique integer n such that a = ϖⁿu with u a unit, and by sending 0 to ∞. If q is the cardinality of the residue field, the absolute value on F induced by its structure as a local field is given by^[5]

$|a|=q^{-v(a)}.$

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 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.
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:

$v\left(\sum_{i=-m}^\infty a_iT^i\right) = -m$ (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.

Higher unit groups

The n^th higher unit group of a non-archimedean local field F is

$U^{(n)}=1%2B\mathfrak{m}^n=\left\{u\in\mathcal{O}^\times:u\equiv1\, (\mathrm{mod}\,\mathfrak{m}^n)\right\}$

for n ≥ 1. The group U⁽¹⁾ is called the group of principal units, and any element of it is called a principal unit. The full unit group $\mathcal{O}^\times$ is denoted U⁽⁰⁾.

The higher unit groups provide a decreasing filtration of the unit group

$\mathcal{O}^\times\supseteq U^{(1)}\supseteq U^{(2)}\supseteq\cdots$

whose quotients are given by

$\mathcal{O}^\times/U^{(n)}\cong\left(\mathcal{O}/\mathfrak{m}^n\right)^\times\text{ and }\,U^{(n)}/U^{(n%2B1)}\approx\mathcal{O}/\mathfrak{m}$

for n ≥ 1.^[6] (Here " $\approx$ " means a non-canonical isomorphism.)

Structure of the unit group

The multiplicative group of non-zero elements of a non-archimedean local field F is isomorphic to

$F^\times\cong(\varpi)\times\mu_{q-1}\times U^{(1)}$

where q is the order of the residue field, and μ_q−1 is the group of (q−1)st roots of unity (in F). Its structure as an abelian group depends on its characteristic:

If F has positive characteristic p, then

$F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/{(q-1)}\oplus\mathbf{Z}_p^\mathbf{N}$

where N denotes the natural numbers;

If F has characteristic zero (i.e. it is a finite extension of Q_p of degree d), then

$F^\times\cong\mathbf{Z}\oplus\mathbf{Z}/(q-1)\oplus\mathbf{Z}/p^a\oplus\mathbf{Z}_p^d$

where a ≥ 0 is defined so that the group of p-power roots of unity in F is $\mu_{p^a}$ .^[7]

Higher dimensional local fields

Main article: Higher local field

It is natural to introduce non-archimedean local fields in a uniform geometric way as the field of fractions of the completion of the local ring of a one-dimensional arithmetic scheme of rank 1 at its non-singular point. For generalizations, a local field is sometimes called a one-dimensional local field.

For a non-negative integer n, an n-dimensional local field is a complete discrete valuation field whose residue field is an (n − 1)-dimensional local field.^[8] Depending on the definition of local field, a zero-dimensional local field is then either a finite field (with the definition used in this article), or a quasi-finite field,^[9] or a perfect field.

From the geometric point of view, n-dimensional local fields with last finite residue field are naturally associated to a complete flag of subschemes of an n-dimensional arithmetic scheme.

Notes

^ Page 20 of Weil 1995
^ See, for example, definition 1.4.6 of Fesenko & Vostokov 2002
^ Page 4 of Weil 1995
^ Corollary 1, page 5 of Weil 1995
^ Weil 1995, chapter I, theorem 6
^ Neukirch 1999, p. 122
^ Neukirch 1999, theorem II.5.7
^ Definition 1.4.6 of Fesenko & Vostokov 2002
^ Serre 1995

References

Serre, Jean-Pierre (1995), Local Fields, Graduate texts in mathematics, 67, Berlin, Heidelberg: 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
Fesenko, Ivan B.; Vostokov, Sergei V. (2002), Local fields and their extensions, Translations of Mathematical Monographs, 121 (Second ed.), Providence, RI: American Mathematical Society, ISBN 9780821832592, MR 1915966
Neukirch, Jürgen (1999), Algebraic Number Theory, Grundlehren der mathematischen Wissenschaften, 322, Berlin: Springer-Verlag, ISBN 978-3-540-65399-8, MR 1697859