Completion of Valued Field

Theorem

Let $\struct {k, \norm {\,\cdot\,} }$ be a valued field.

Then there exists a completion $\struct {k', \norm {\,\cdot\,}'}$ of $\struct {k, \norm {\,\cdot\,} }$ such that $\struct {k', \norm {\,\cdot\,}'}$ is a valued field.

Furthermore, every completion of $\struct{k, \norm {\,\cdot\,} }$ is isometrically isomorphic to $\struct {k', \norm {\,\cdot\,}'}$.

Proof

By Completion of Normed Division Ring then $\struct {k, \norm {\, \cdot \,} }$ has a normed division ring completion $\struct {k', \norm {\, \cdot \,}'}$

By Normed Division Ring is Field iff Completion is Field then $\struct {k', \norm {\, \cdot \,}'}$ is a field.

By Normed Division Ring Completions are Isometric and Isomorphic then every completion of $\struct {k, \norm {\,\cdot\,} }$ is isometrically isomorphic to $\struct {k', \norm {\,\cdot\,}'}$.

$\blacksquare$

Examples

• The completion of $\Q$ with respect to the usual absolute value is $\R$
• The completion of $\Q$ with respect to the $p$-adic norm is known as the field of $p$-adic numbers and denoted $\Q_p$
• By Ostrowski's Theorem there are no other completions of $\Q$ as a valued field.