Field Q_3 - Knowlpedia

The $3$ -adic number field $\mathbb Q_3$ is the completion of $\mathbb Q$ for $|x|_3=3^{-v_3(x)}$ , where $v_3$ is the $3$ -adic valuation. It carries the topology from this absolute value.

Integral elements and topology

The valuation ring is $\mathbb Z_3$ . Its maximal ideal is $3\mathbb Z_3$ , its residue field is $\mathbb F_3$ , and $3$ is a uniformizer. The field is locally compact, complete, nondiscrete and totally disconnected.

Category distinctions

As a rational algebra it carries a specified inclusion of $\mathbb Q$ , but it has infinite rational dimension. Topological-field morphisms must be continuous as well as preserve field operations. It has characteristic zero; the characteristic of its residue field is a different invariant.

References

J. S. Milne, Algebraic Number Theory, Chapter 7, Theorem 7.23; Propositions 7.26, 7.46; Remark 7.49, pp. 115–127.