Algebras and modules
Model category presentations
Geometry on formal duals of algebras
In field theory, what we call an ‘absolute value’ here is often called a ‘valuation’. However, there is also a more general notion of valuation used in field theory, which is what we call ‘valuation’. The notion of absolute value is also used in functional analysis, where it may be called a ‘multiplicative norm’ (rather than merely submultiplicative, as norms on Banach algebras are required to be).
For a rig (typically either a field or at least an integral domain, or else an associative algebra over such), an absolute value on is a (non-trivial) multiplicative seminorm, or equivalently a finite real-valued valuation.
This means it is a function
to the real numbers such that for all
precisely if ;
(the triangle inequality).
If the last trinagle inequality is strengthened to
then is called an ultrametric or non-archimedean absolute value. Otherwise it is called archimedean.
Two absolute values and are called equivalent if for all
An equivalence class of absolute values is also called a place.
A field equipped with an absolute value which is a complete metric space with respect to the corresponding metric is called a complete field.
Trivial absolute value
Every field admits the trivial absolute value defined by
This is non-archimedean.
On the real and complex numbers
The standard absolute value on the real numbers is
The standard absolute value on the complex numbers is
These standard absolute values are archimedean, and with respect to these standard absolute values, both and are complete and hence are complete archimedean valued fields. Notice that is in addition an ordered field and as such also an archimedean field.
On the rational numbers
The standard absolute value above restricts to the standard absolute value on the rational numbers
Moreover, for any prime number there is a (class of an) absolute value on defined by
for any and where is the unique such decomposition of a given rational number with not divisible by .
This is called the -adic absolute value. It is non-archimedean. The completion of by this is called the field of p-adic numbers, which is therefore a non-archimedean field.
Ostrowski's theorem says that this exhaust the non-trivial absolute values on the rational numbers. Therefore the real numbers and the p-adic numbers are the only possible completions of .
On Laurent power series
The field of Laurent series over a field is a complete field with respect to the absolute value that sends a series to for a fixed and with the lowest integer such that the th coefficient of the series is not .
Section 1.5, 1.6 of