Norm on a field

$ K $

A mapping $ \phi $ from $ K $ to the set $ \mathbf R $ of real numbers, which satisfies the following conditions:

1) $ \phi ( x) \geq 0 $, and $ \phi ( x) = 0 $ if and only if $ x = 0 $;

2) $ \phi ( x \cdot y ) = \phi ( x ) \cdot \phi ( y ) $;

3) $ \phi ( x + y ) \leq \phi ( x ) + \phi ( y ) $.

Hence $ \phi ( 1) = \phi ( - 1 ) = 1 $; $ \phi ( x ^ {-1} ) = \phi( x) ^ {-1} $.

The norm of $ x $ is often denoted by $ | x | $ instead of $ \phi ( x) $. A norm is also called an absolute value or a multiplicative valuation. Norms may (more generally) be considered on any ring with values in a linearly ordered ring [4]. See also Valuation.

Examples of norms. If $ K = \mathbf R $, the field of real numbers, then $ | x | = \max \{ x, - x \} $, the ordinary absolute value or modulus of the number $ x \in \mathbf R $, is a norm. Similarly, if $ K $ is the field $ \mathbf C $ of complex numbers or the skew-field $ \mathbf H $ of quaternions, then $ | x | = \sqrt {x \cdot \overline{x}\; } $ is a norm. The subfields of these fields are thus also provided with an induced norm. Any field has the trivial norm:

$$ \phi ( x ) = \left \{ \begin{array}{cl} 0, & x = 0; \\ 1, & x \neq 0. \\ \end{array} \right .$$

Finite fields and their algebraic extensions only have the trivial norm.

Examples of norms of another type are provided by logarithmic valuations of a field $ K $: If $ v $ is a valuation on $ K $ with values in the group $ \mathbf R $ and if $ a $ is a real number, $ 0 < a < 1 $, then $ \phi ( x) = a ^ {v( x)} $ is a norm. For example, if $ K = \mathbf Q $ and $ v _ {p} $ is the $ p $- adic valuation of the field $ \mathbf Q $, then $ {| x | } _ {p} = ( 1/p) ^ {v _ {p} ( x) } $ is called the $ p $-adic absolute value or the $ p $-adic norm. These absolute values satisfy the following condition, which is stronger than 3):

4) $ \phi ( x + y ) \leq \max \{ \phi ( x ) , \phi ( y ) \} $.

Norms satisfying condition 4) are known as ultra-metric norms or non-Archimedean norms (as distinct from Archimedean norms which do not satisfy this condition (but do satisfy 3)). They are distinguished by the fact that $ \phi ( n \cdot 1) \leq 1 $ for all integers $ n $. All norms on a field of characteristic $ p > 0 $ are ultra-metric. All ultra-metric norms are obtained from valuations as indicated above: $ \phi = a ^ {v( x)} $ (and conversely, $ - \mathop{\rm log} \phi $ can always be taken as a valuation).

A norm $ \phi $ defines a metric on $ K $ if $ \phi ( x - y) $ is taken as the distance between $ x $ and $ y $, and in this way it defines a topology on $ K $. The topology of any locally compact field is defined by some norm. Two norms $ \phi _ {1} $ and $ \phi _ {2} $ are said to be equivalent if they define the same topology; in a such case there exists a $ \lambda > 0 $ such that $ \phi _ {1} ( x) = \phi _ {2} ( x) ^ \lambda $ for all $ x \in K $.

The structure of all Archimedean norms is given by Ostrowski's theorem: If $ \phi $ is an Archimedean norm on a field $ K $, then there exists an isomorphism of $ K $ into a certain everywhere-dense subfield of one of the fields $ \mathbf R $, $ \mathbf C $ or $ \mathbf H $ such that $ \phi $ is equivalent to the norm induced by that of $ \mathbf R $, $ \mathbf C $ or $ \mathbf H $.

Any non-trivial norm of the field $ \mathbf Q $ of rational numbers is equivalent either to a $ p $- adic norm $ {| \cdot | } _ {p} $, where $ p $ is a prime number, or to the ordinary norm. For any rational number $ r \in \mathbf Q $ one has

$$ | r | \prod _ { p } | r | _ {p} = 1. $$

A similar formula is also valid for algebraic number fields [2], [3].

If $ \phi $ is a norm on a field $ K $, then $ K $ may be imbedded by the classical completion process in a field $ K _ \phi $ that is complete with respect to the norm that (uniquely) extends $ \phi $( cf. Complete topological space). One of the principal modern methods in the study of fields is the imbedding of a field $ K $ into the direct product $ \prod _ \phi K _ \phi $ of all completions $ K _ \phi $ of the field $ K $ with respect to all non-trivial norms of $ K $( see Adèle). If $ K $ admits non-trivial valuations, then it is dense in $ \prod _ \phi K _ \phi $ in the adèlic topology; in fact, if $ \phi _ {1} \dots \phi _ {n} $ are non-trivial, non-equivalent norms on $ K $, if $ a _ {1} \dots a _ {n} $ are elements of $ K $ and if $ \epsilon > 0 $, then there exists an $ a \in K $ such that $ \phi _ {i} ( a - a _ {i} ) < \epsilon $ for all $ i $( the approximation theorem for norms).

A norm on a field $ K $ may be extended (in general, non-uniquely) to any algebraic field extension of the field $ K $. If $ K $ is complete with respect to the norm $ \phi $ and if $ L $ is an extension of $ K $ of degree $ n $, the extension of $ \phi $ to $ L $ is unique, and is given by the formula

$$ \phi ^ \prime ( x ) = \{ \phi ( N _ {L / K } ( x ) ) \} ^ {1 / n } $$

for $ x \in L $.

References

[1]	N. Bourbaki, "Elements of mathematics. Commutative algebra" , Addison-Wesley (1972) (Translated from French)
[2]	Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
[3]	S. Lang, "Algebra" , Addison-Wesley (1984)
[4]	A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
[5]	J.W.S. Cassels (ed.) A. Fröhlich (ed.) , Algebraic number theory , Acad. Press (1986)

Comments

Non-Archimedean norms satisfy $ \phi ( n \cdot 1 ) \leq \phi ( 1) $ and hence do not satisfy the Archimedean axiom, whence the appellation.