Non-Archimedean Valuation

Let K be a field of arbitrary characteristic. Let v:K->R union {infty} be defined by the following properties:

1. v(x)=infty<=>x=0,

2. v(xy)=v(x)+v(y)  forall x,y in K, and

3. v(x+y)>=inf{v(x),v(y)}.

Then v is said to be a non-Archimedean valuation and K is said to be a non-Archimedean valued field. For example, for K=Q, if x in Q^*, x can be decomposed as x=p^tx_0 where t in Z and neither the numerator nor denominator of x_0 involve p. Then v(x)=v_p(x)=t is a non-Archimedean valuation.

Another example can be constructed by letting K=F_r(1/T) be the field of formal Laurent series over the finite field with r elements F_r, taking k=F_r(T) the quotient field of F_r[T] (the polynomial ring in the variable T over F_r), and setting v(0)=infty. If x in k^* is written (1/T)^ax_0 with the numerator and denominator of x_0 relatively prime to T, then v(x)=a is a non-Archimedean valuation.

Let v be a non-Archimedean valuation and let alpha in R with 0<alpha<1. The non-Archimedean absolute value |·|_v:K->[0,+infty) is obtained by setting |x|_v=alpha^(v(x)).

|·|_v has the following properties:

1. |x|_v=0<=>x=0

2. |xy|_v=|x|_v|y|_v

3. |x+y|_v<=max{|x|_v,|y|_v} (non-Archimedean triangular inequality).

An absolute value |·| over a field K is non-Archimedean iff |n1|<=1 for all n in Z.

The completion K^^ of K is the completion of the metric space (K,|·|). In the example above, Q_p is the completion of Q with respect to the valuation v_p.

Non-Archimedean complete fields satisfy the following properties:

1. sum_(n)a_n converges <=> lima_n=0. (Note that for Archimedean valuations we only have => implication.)

2. If a_n!=0  forall n, then product_(n)a_n converges to a nonzero element if and only if lima_n=1.


 R=R_K:={x in K:|x|_v<=1},

so R is the valuation ring of K and

 M=M_K={x in K:|x|_v<1}.

Then R is a local ring and M is its maximal ideal. If K is a non-Archimedean complete field, then R is compact and R/M is a finite field. Suppose that the cardinality of R/M is q, then the definition alpha=1/q can be made and the absolute value is said to be normalized.

See also

Krasner's Lemma, Non-Archimedean Field, Non-Archimedean Geometry

This entry contributed by José Gallardo Alberni

Explore with Wolfram|Alpha


López, B. "Analytic Theory of Drinfeld Modules." Proceedings of the Workshop on Drinfeld Modules, Modular Schemes and Applications, Alden-Biesen, 9-14 September 1996 (Ed. E.-U. Gekeler, M. van der Put, M. Reversat, and J. Van Geel). Singapore: World Scientific, pp. 32-33, 1997.Goss, D. Basic Structures of Function Field Arithmetic. Berlin: Springer-Verlag, pp. 35-45, 1996.Jacobson, N. Basic Algebra II, 2nd ed. New York: W. H. Freeman, pp. 557-618, 1989.

Referenced on Wolfram|Alpha

Non-Archimedean Valuation

Cite this as:

Alberni, José Gallardo. "Non-Archimedean Valuation." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications