# 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)

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