Completion of a valued field

Remarks

A completion exists and is unique up to a unique isomorphism of fields that preserves the absolute values and commutes with the maps from $K$. One construction: the Cauchy sequences in $K$ form a ring under termwise operations. The sequences tending to $0$ form a maximal ideal, and the quotient is a field. The absolute value of the class of $(x_n)$ is $\lim_n \abs{x_n}$. As a metric space it is the completion of the metric space $K$. Example: $\R$ with the usual absolute value and the inclusion $\Q \to \R$ is a completion of $\Q$; clause 3 holds because every real number is a limit of rationals. Example: for a prime $p$, the p-adic numbers $\Q_p$ are the completion of $\Q$ for the p-adic absolute value. Non-example: the inclusion $\Q \to \C$ satisfies clauses 1 and 2 for the usual absolute values but fails clause 3, since the closure of $\Q$ in $\C$ is $\R$. For a non-Archimedean absolute value, the completion has the same value group and the same residue field as $K$.