Norm on a field

A mapping from to the set of real numbers, which satisfies the following conditions:

1) , and if and only if ;

2) ;

3) .

Hence ; .

The norm of is often denoted by instead of . 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 , the field of real numbers, then , the ordinary absolute value or modulus of the number , is a norm. Similarly, if is the field of complex numbers or the skew-field of quaternions, then is a norm. The subfields of these fields are thus also provided with an induced norm. Any field has the trivial norm:

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 : If is a valuation on with values in the group and if is a real number, , then is a norm. For example, if and is the -adic valuation of the field , then is called the -adic absolute value or the -adic norm. These absolute values satisfy the following condition, which is stronger than 3):

4) .

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 for all integers . All norms on a field of characteristic are ultra-metric. All ultra-metric norms are obtained from valuations as indicated above: (and conversely, can always be taken as a valuation).

A norm defines a metric on if is taken as the distance between and , and in this way it defines a topology on . The topology of any locally compact field is defined by some norm. Two norms and are said to be equivalent if they define the same topology; in a such case there exists a such that for all .

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

Any non-trivial norm of the field of rational numbers is equivalent either to a -adic norm , where is a prime number, or to the ordinary norm. For any rational number one has

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

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

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

for .

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 and hence do not satisfy the Archimedean axiom, whence the appellation.