Completion of Valued Field

From ProofWiki
Jump to: navigation, search

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.